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

List of thumbnails

< >
181
181 (161) 		182
182 (162)
183
183 (163) 		184
184 (164)
185
185 (165) 		186
186 (166)
187
187 (167) 		188
188 (168)
189
189 (169) 		190
190 (170)
< >
page |< < (163) of 677 > >|
    <echo version="1.0RC">
      <text xml:lang="it" type="free">
        <div xml:id="echoid-div556" type="section" level="1" n="145">
          <p>
            <s xml:id="echoid-s10502" xml:space="preserve">
              <pb o="163" file="0183" n="183" rhead="LIBER SECVNDVS."/>
            in eodem horologio, qui angulo D H F, æqualis eſt. </s>
            <s xml:id="echoid-s10503" xml:space="preserve">Quod hac ratione oſtendemus. </s>
            <s xml:id="echoid-s10504" xml:space="preserve">Ducta recta
              <lb/>
            D K, erit D K, ipſi D F, æqualis; </s>
            <s xml:id="echoid-s10505" xml:space="preserve">propterea quòd latera I D, I F, trianguli D I F, lateribus I D, I K,
              <lb/>
              <note position="right" xlink:label="note-0183-01" xlink:href="note-0183-01a" xml:space="preserve">4. primi.</note>
            trianguli D I K, æqualia ſint, angulosq́; </s>
            <s xml:id="echoid-s10506" xml:space="preserve">contineant æquales, nimirum rectos. </s>
            <s xml:id="echoid-s10507" xml:space="preserve">Quoniam igitur la-
              <lb/>
            tera D H, D F, trianguli D H F, in horologio, lateribus D H, D k, trianguli D H k, in eodem horo-
              <lb/>
            logio æqualia ſunt, angulosq́; </s>
            <s xml:id="echoid-s10508" xml:space="preserve">continent æquales, vtpote rectos; </s>
            <s xml:id="echoid-s10509" xml:space="preserve">Eſt enim axis H D, rectus exiſtens
              <lb/>
            ad planum Aequatoris, ad rectas D F, D K, in plano eodem Aequatoris exiſtentes perpendicularis,
              <lb/>
            ex defin. </s>
            <s xml:id="echoid-s10510" xml:space="preserve">3. </s>
            <s xml:id="echoid-s10511" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10512" xml:space="preserve">11. </s>
            <s xml:id="echoid-s10513" xml:space="preserve">Euclidis; </s>
            <s xml:id="echoid-s10514" xml:space="preserve">æquales erunt anguli D H F, D H k) erit quoque reliquus X H D, in
              <lb/>
            dicta figura reliquo D H X, in horologio æqualis; </s>
            <s xml:id="echoid-s10515" xml:space="preserve">& </s>
            <s xml:id="echoid-s10516" xml:space="preserve">ſic de aliis. </s>
            <s xml:id="echoid-s10517" xml:space="preserve">Quæ cum ita ſint, coniungetur
              <lb/>
            recta H X, dictæ figuræ cum recta H X, horologii, in illa circumuolutione radiorum, propter an-
              <lb/>
            gulo rum æqualitatem, quos rectę H X, H X, faciunt cum axe H D, &</s>
            <s xml:id="echoid-s10518" xml:space="preserve">c. </s>
            <s xml:id="echoid-s10519" xml:space="preserve">Eademq́; </s>
            <s xml:id="echoid-s10520" xml:space="preserve">eſt ratio de cæte-
              <lb/>
              <note position="left" xlink:label="note-0183-02" xlink:href="note-0183-02a" xml:space="preserve">10</note>
            ris. </s>
            <s xml:id="echoid-s10521" xml:space="preserve">Conſtat igitur Ioan. </s>
            <s xml:id="echoid-s10522" xml:space="preserve">Baptiſtam Benedictum in ſua Gnomonica immerito deſcriptionẽ hanc
              <lb/>
            arcuum ſignorum reprehendere.</s>
            <s xml:id="echoid-s10523" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10524" xml:space="preserve">PORRO deſcriptis hyperbolis borealium ſignorum, hoc eſt, quæ inter centrum H, & </s>
            <s xml:id="echoid-s10525" xml:space="preserve">æqui-
              <lb/>
              <note position="right" xlink:label="note-0183-03" xlink:href="note-0183-03a" xml:space="preserve">Quomodo ex
                <lb/>
              hyperbolis ſi-
                <lb/>
              gnorum borea-
                <lb/>
              lium deſcriban
                <lb/>
              tur hyperbolæ
                <lb/>
              auſtraliũ ſign@@
                <lb/>
              ium.</note>
            noctialem lineam continentur, deſcribemus accuratius hyperbolas oppoſitas ſignorum auſtraliũ,
              <lb/>
            id eſt, quæ ex altera parte lineæ æquinoctialis deſcribuntur, (quoniam hæ difficilius deſcribentur,
              <lb/>
            quòd puncta in lineis horarijs vltra lineam æquinoctialem, per quæ ducendæ ſunt, magis inter ſe
              <lb/>
            diſtent, quàm citra lineam æquinoctialem) hac ratione. </s>
            <s xml:id="echoid-s10526" xml:space="preserve">Inuenta diametro transuerſa oppoſita-
              <lb/>
            rum ſectionum in linea meridiana horologij, quæ quidem æqualis ſemper eſt portioni rectę H B,
              <lb/>
            in figura radiorum inter radios ſignorum oppoſitorum interceptæ, (quemadmodum in horolo-
              <lb/>
            gio recta K N, diameter eſt oppoſitarum ſectionum ♋, & </s>
            <s xml:id="echoid-s10527" xml:space="preserve">♑, atque ęqualis portioni μ a, rectæ
              <lb/>
              <note position="left" xlink:label="note-0183-04" xlink:href="note-0183-04a" xml:space="preserve">20</note>
            H B, in figura radiorum inter radios ♋, & </s>
            <s xml:id="echoid-s10528" xml:space="preserve">♑, interiectæ) diuidemus eam bifariam, vt habeamus
              <lb/>
            centrum oppoſitarum ſectionum, ſecundum doctrinam Apollonii in ſecundis definitionibus lib.
              <lb/>
            </s>
            <s xml:id="echoid-s10529" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10530" xml:space="preserve">conicorum elementorum. </s>
            <s xml:id="echoid-s10531" xml:space="preserve">Deinde quia per propoſ. </s>
            <s xml:id="echoid-s10532" xml:space="preserve">30. </s>
            <s xml:id="echoid-s10533" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10534" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10535" xml:space="preserve">Apollonii, recta linea quæcunque
              <lb/>
            per centrum oppoſitarum ſectionum ducta in centro ſecatur bifariam, ducemus ex punctis linea-
              <lb/>
            rum horariarum ſupra lineam ęquinoctialem, per quæ hyperbolæ boreales tranſeunt, per centrũ
              <lb/>
            inuentum lineas occultas. </s>
            <s xml:id="echoid-s10536" xml:space="preserve">Si enim ſegmentis illarum inter dicta puncta, & </s>
            <s xml:id="echoid-s10537" xml:space="preserve">centrum poſitis ab-
              <lb/>
            ſcindamus infra centrum dictum lineas ęquales, habebimus in lineis illis occultis puncta, per quæ
              <lb/>
            hyperbolæ auſtrales ducendæ ſunt. </s>
            <s xml:id="echoid-s10538" xml:space="preserve">Qua arte, & </s>
            <s xml:id="echoid-s10539" xml:space="preserve">induſtria vtemur quoque in ſequentibus horolo-
              <lb/>
            giis, in quibus oppoſitæ hyperbolæ deſcribendę erunt, ſiue illæ ſint parallelorum Æquatoris per
              <lb/>
            initia ſignorum Zodiaci ductorum, ſiue parallelorum Horizontis.</s>
            <s xml:id="echoid-s10540" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">30</note>
          <p>
            <s xml:id="echoid-s10541" xml:space="preserve">HÆC ratio deſcribendarum hyperbolarum auſtralium ſignorum ex hyperbolis ſignorũ bo-
              <lb/>
            realium planius intelligetur ex ſequenti figura: </s>
            <s xml:id="echoid-s10542" xml:space="preserve">In qua diameter oppoſitarum hyperbolarum eſt
              <lb/>
            D E, & </s>
            <s xml:id="echoid-s10543" xml:space="preserve">centrum earum punctum F. </s>
            <s xml:id="echoid-s10544" xml:space="preserve">Si igitur ex puncto K, ſuperioris hyperbolæ ducatur per cen-
              <lb/>
            trum F, recta K F N, abſcindaturq́; </s>
            <s xml:id="echoid-s10545" xml:space="preserve">F N, ipſi F K, æqualis, ducenda erit per punctum N, hyperbola
              <lb/>
            oppoſita; </s>
            <s xml:id="echoid-s10546" xml:space="preserve">quandoquidem ex propoſ. </s>
            <s xml:id="echoid-s10547" xml:space="preserve">30. </s>
            <s xml:id="echoid-s10548" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10549" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10550" xml:space="preserve">Apolì. </s>
            <s xml:id="echoid-s10551" xml:space="preserve">recta F k, æqualis eſt ſegmento eiuſdem re-
              <lb/>
            ctæ vltra F, extenſæ inter F, centrum & </s>
            <s xml:id="echoid-s10552" xml:space="preserve">oppoſitam hyperbolam comprehenſo. </s>
            <s xml:id="echoid-s10553" xml:space="preserve">Sic etiam, ducta
              <lb/>
            recta L F M, ſi rectæ F L, abſcindatur recta F M, ducenda erit oppoſita hyperbole per punctum M;
              <lb/>
            </s>
            <s xml:id="echoid-s10554" xml:space="preserve">& </s>
            <s xml:id="echoid-s10555" xml:space="preserve">ſic de cæteris. </s>
            <s xml:id="echoid-s10556" xml:space="preserve">Ducendæ porro erunt, meo iudicio, rectæ per centrum hyperbolarum oppoſita-
              <lb/>
            rum ex illis punctis borealium hyperbolarum, per quæ tranſeunt lineæ horariæ: </s>
            <s xml:id="echoid-s10557" xml:space="preserve">quoniã illa pun-
              <lb/>
            cta per conſtructionem ſunt inuenta. </s>
            <s xml:id="echoid-s10558" xml:space="preserve">Vnde accuratius per illa deſcribemus hyperbolam oppoſi-
              <lb/>
              <note position="left" xlink:label="note-0183-06" xlink:href="note-0183-06a" xml:space="preserve">40</note>
            tam, quàm per alia puncta inter illa intermedia, quæ non ſunt per conſtructionem inuenta, ſed
              <lb/>
            per coniecturam.</s>
            <s xml:id="echoid-s10559" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10560" xml:space="preserve">FACILE autem ex propoſ. </s>
            <s xml:id="echoid-s10561" xml:space="preserve">6. </s>
            <s xml:id="echoid-s10562" xml:space="preserve">ſuperioris lib. </s>
            <s xml:id="echoid-s10563" xml:space="preserve">cognoſcemus, quinam paralleli faciant in ho-
              <lb/>
            rologio ſectiones oppoſitas, hoc eſt, hyperbolas, vel alias ſectiones. </s>
            <s xml:id="echoid-s10564" xml:space="preserve">In ſolas enim hyperbolas qua-
              <lb/>
            drat prædicta ratio. </s>
            <s xml:id="echoid-s10565" xml:space="preserve">Quod tamen etiam ex figura radiorum Zodiaci paulo ante deſcripta ita eli-
              <lb/>
              <note position="right" xlink:label="note-0183-07" xlink:href="note-0183-07a" xml:space="preserve">Quomodo co-
                <lb/>
              gnoſcatur, an a@
                <lb/>
              cus ſignorũ ſint
                <lb/>
              hyper@olæ, pa-
                <lb/>
              rabolæ, aut elli-
                <lb/>
              pſes.</note>
            ciemus. </s>
            <s xml:id="echoid-s10566" xml:space="preserve">Quotieſcunque recta H B, in dicta figura ſecat duos radios ſignorum oppoſitorum, hoc
              <lb/>
            eſt, radios æqualiter hinc inde à radio Æquatoris diſtantes, quales ſunt radij ♋, & </s>
            <s xml:id="echoid-s10567" xml:space="preserve">♑; </s>
            <s xml:id="echoid-s10568" xml:space="preserve">♊, & </s>
            <s xml:id="echoid-s10569" xml:space="preserve">♐; </s>
            <s xml:id="echoid-s10570" xml:space="preserve">♉,
              <lb/>
            & </s>
            <s xml:id="echoid-s10571" xml:space="preserve">♏, &</s>
            <s xml:id="echoid-s10572" xml:space="preserve">c. </s>
            <s xml:id="echoid-s10573" xml:space="preserve">arcus, ſeu paralleli illorum oppoſitorum ſignorum ſunt hyperbolæ oppoſitæ. </s>
            <s xml:id="echoid-s10574" xml:space="preserve">Quoniam
              <lb/>
            enim in triangulo D H a, angulus externus A D a, maior eſt interno oppoſito D H a; </s>
            <s xml:id="echoid-s10575" xml:space="preserve">eſt autem an-
              <lb/>
              <note position="right" xlink:label="note-0183-08" xlink:href="note-0183-08a" xml:space="preserve">16. primi.</note>
            gulus A D a, complemento declinationis paralleli, cuius radius D a, atque adeò & </s>
            <s xml:id="echoid-s10576" xml:space="preserve">oppoſiti, cuius
              <lb/>
              <note position="left" xlink:label="note-0183-09" xlink:href="note-0183-09a" xml:space="preserve">50</note>
            radius D μ, æqualis; </s>
            <s xml:id="echoid-s10577" xml:space="preserve">angulus autem D H a, altitudini poli ſupra Horizontem æqualis eſt, ſecabit
              <lb/>
            Horizon vtrumque parallelum radiorũ D a, D μ, vt in coroll propoſ. </s>
            <s xml:id="echoid-s10578" xml:space="preserve">6.</s>
            <s xml:id="echoid-s10579" xml:space="preserve">
              <unsure/>
            ſuperioris lib. </s>
            <s xml:id="echoid-s10580" xml:space="preserve">docuimus.
              <lb/>
            </s>
            <s xml:id="echoid-s10581" xml:space="preserve">Quare per eandem@propoſ. </s>
            <s xml:id="echoid-s10582" xml:space="preserve">6. </s>
            <s xml:id="echoid-s10583" xml:space="preserve">arcus illorum ſignorum hyperbolæ ſunt oppoſitæ, & </s>
            <s xml:id="echoid-s10584" xml:space="preserve">ęquales. </s>
            <s xml:id="echoid-s10585" xml:space="preserve">Quan-
              <lb/>
            do verò recta H B, non ſecat vtrumque radium ſignorum oppoſitorum, ſed vni æquidiſtat, & </s>
            <s xml:id="echoid-s10586" xml:space="preserve">alte-
              <lb/>
            rum ſecat; </s>
            <s xml:id="echoid-s10587" xml:space="preserve">erit arcus illius ſigni, cuius radius ſecatur, Parabola. </s>
            <s xml:id="echoid-s10588" xml:space="preserve">Quoniam enim tunc externus
              <lb/>
            angulus A D a, cõplementi declinationis, æqualis eſt interno angulo D H μ, altitudinis poli, quòd
              <lb/>
              <note position="right" xlink:label="note-0183-10" xlink:href="note-0183-10a" xml:space="preserve">29. primi.</note>
            H μ, D a, parallelæ ponantur; </s>
            <s xml:id="echoid-s10589" xml:space="preserve">tanget Horizon, ex coroll. </s>
            <s xml:id="echoid-s10590" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s10591" xml:space="preserve">5. </s>
            <s xml:id="echoid-s10592" xml:space="preserve">antecedentis lib. </s>
            <s xml:id="echoid-s10593" xml:space="preserve">vtrumque pa-
              <lb/>
            rallelorum oppoſitorum. </s>
            <s xml:id="echoid-s10594" xml:space="preserve">Quare per eandem propoſ. </s>
            <s xml:id="echoid-s10595" xml:space="preserve">5. </s>
            <s xml:id="echoid-s10596" xml:space="preserve">arcus alterius, cuius radius ſecatur, Para-
              <lb/>
            bole erit. </s>
            <s xml:id="echoid-s10597" xml:space="preserve">Quando denique recta H B, neque vtrumque radium oppoſitorum ſignorum ſecat, neq;
              <lb/>
            </s>
            <s xml:id="echoid-s10598" xml:space="preserve">vni æquidiſtat, ſed vnum quidem ſecat, ab altero autem ſemper magis, ac magis recedit, erit arcus
              <lb/>
            illius ſigni, cuius radius ſecatur, Ellipſis. </s>
            <s xml:id="echoid-s10599" xml:space="preserve">Nam quia tunc angulus A D a, complementi </s>
          </p>
        </div>
      </text>
    </echo>
Impressum