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

Page concordance

< >
Scan Original
181 161
182 162
183 163
184 164
185 165
186 166
187 167
188 168
189 169
190 170
191 171
192 172
193 173
194 174
195 175
196 176
197 177
198 178
199 179
200 180
201 181
202 182
203 183
204 184
205 185
206 186
207 187
208 188
209 189
210 190
< >
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