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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        <div xml:id="echoid-div105" type="section" level="1" n="33">
          <p style="it">
            <s xml:id="echoid-s2033" xml:space="preserve">
              <pb o="30" file="0050" n="50" rhead="GNOMONICES"/>
            gulum B C, ſub lateribus trianguli per axem comprehenſum; </s>
            <s xml:id="echoid-s2034" xml:space="preserve">erit, vt quadratum baſis BC,
              <lb/>
            ad rectangulum ſub lateribus A B, A C, contentum, ita E K, ad A E. </s>
            <s xml:id="echoid-s2035" xml:space="preserve">Quare ex propoſ.
              <lb/>
            </s>
            <s xml:id="echoid-s2036" xml:space="preserve">11. </s>
            <s xml:id="echoid-s2037" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2038" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2039" xml:space="preserve">Apollonij, E K, latus rectum eſt paraboles E F G, hoc eſt, Recta, iuxta quam poſ-
              <lb/>
            ſunt ordinatim applicatæ, & </s>
            <s xml:id="echoid-s2040" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2041" xml:space="preserve"/>
          </p>
          <figure number="32">
            <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0050-01"/>
          </figure>
          <note position="left" xml:space="preserve">10</note>
          <figure number="33">
            <image file="0050-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0050-02"/>
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          <note position="left" xml:space="preserve">20</note>
          <p>
            <s xml:id="echoid-s2042" xml:space="preserve">INVENTO igitur latere recto, ſumatur in plano aliquo axis parabolæ quicunque E H. </s>
            <s xml:id="echoid-s2043" xml:space="preserve">(De
              <lb/>
              <note position="left" xlink:label="note-0050-03" xlink:href="note-0050-03a" xml:space="preserve">Alia deſ@riptio
                <lb/>
              Paraboles in
                <lb/>
              plano.</note>
            illa enim Parabola hic agimus, cuius diameter etiam axis eſt, ſecans omnes ordinatim applicatas bifa-
              <lb/>
            riam, & </s>
            <s xml:id="echoid-s2044" xml:space="preserve">ad rectos angulos) in quo ſumantur quot cunque partes inter ſe æquales, (quò autem minores hæ
              <lb/>
            partes fuerint, eò accuratius parabola deſcribetur) ita vt E A, ſit 1; </s>
            <s xml:id="echoid-s2045" xml:space="preserve">A B, 3; </s>
            <s xml:id="echoid-s2046" xml:space="preserve">B C, 5; </s>
            <s xml:id="echoid-s2047" xml:space="preserve">C H, 7, & </s>
            <s xml:id="echoid-s2048" xml:space="preserve">ſic
              <lb/>
            deinceps, ſecundum numerorum imparium ſeriem: </s>
            <s xml:id="echoid-s2049" xml:space="preserve">atque per puncta A, B, C, H, &</s>
            <s xml:id="echoid-s2050" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2051" xml:space="preserve">ad E H, perpen-
              <lb/>
            diculares vtrinque ducantur eo modo, quo ſupra docuimus. </s>
            <s xml:id="echoid-s2052" xml:space="preserve">Deinde inter latus rectum E k, & </s>
            <s xml:id="echoid-s2053" xml:space="preserve">rectam
              <lb/>
            E A, inuenta media proportionali, abſcindatur ei vtrinq; </s>
            <s xml:id="echoid-s2054" xml:space="preserve">æqualis A D; </s>
            <s xml:id="echoid-s2055" xml:space="preserve">& </s>
            <s xml:id="echoid-s2056" xml:space="preserve">ex B, vtrinq; </s>
            <s xml:id="echoid-s2057" xml:space="preserve">abſcindatur
              <lb/>
            B F, dupla ipſius A D; </s>
            <s xml:id="echoid-s2058" xml:space="preserve">& </s>
            <s xml:id="echoid-s2059" xml:space="preserve">ex C, vtrinque C G, tripla eiuſdem A D, & </s>
            <s xml:id="echoid-s2060" xml:space="preserve">ex H, ipſa H I, quadrupla, & </s>
            <s xml:id="echoid-s2061" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0050-04" xlink:href="note-0050-04a" xml:space="preserve">30</note>
            ſic deinceps ſecundum naturalem ſeriem numerorum. </s>
            <s xml:id="echoid-s2062" xml:space="preserve">Nam per puncta D, F, G, I, deſcribenda erit para-
              <lb/>
            bola. </s>
            <s xml:id="echoid-s2063" xml:space="preserve">Quod enim per punctum D, tranſeat, ex eo probatur, quod quadratum ex A D, recta, quæ media
              <lb/>
            proportionalis est inter E K, E A, æquale eſt rectangulo ſub E K, E A, atque adeò A D, ordinatim
              <lb/>
              <note position="left" xlink:label="note-0050-05" xlink:href="note-0050-05a" xml:space="preserve">17. ſexti.</note>
            applicata eſt in parabola, cuius latus rectum E K, vt conſtat ex propoſ. </s>
            <s xml:id="echoid-s2064" xml:space="preserve">11. </s>
            <s xml:id="echoid-s2065" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2066" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2067" xml:space="preserve">Apollonij. </s>
            <s xml:id="echoid-s2068" xml:space="preserve">Quare pa-
              <lb/>
            rabola per punctum D, tranſibit. </s>
            <s xml:id="echoid-s2069" xml:space="preserve">Si enim per aliud punctum, vt per P, tranſiret, eſſet quadratum quc
              <unsure/>
            -
              <lb/>
            que ex A P, rectangulo ſub E K, E A, æquale, ex propoſ. </s>
            <s xml:id="echoid-s2070" xml:space="preserve">11. </s>
            <s xml:id="echoid-s2071" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2072" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2073" xml:space="preserve">Apollonij, quòd A P, ordinatim ap-
              <lb/>
            plicata eſſet ad diametrum. </s>
            <s xml:id="echoid-s2074" xml:space="preserve">Quare quadrata ex A D, A P, æqualia, & </s>
            <s xml:id="echoid-s2075" xml:space="preserve">ipſæ rectæ æquales eſſent, pars
              <lb/>
            & </s>
            <s xml:id="echoid-s2076" xml:space="preserve">totum, quod eſt abſurdum. </s>
            <s xml:id="echoid-s2077" xml:space="preserve">Tranſit ergo parabola, cuius latus rectum E K, per D, punctum. </s>
            <s xml:id="echoid-s2078" xml:space="preserve">Quod
              <lb/>
            autem tranſeat quoque per puncta F, G, I, ita oſtendemus. </s>
            <s xml:id="echoid-s2079" xml:space="preserve">Quoniam recta B F, dupla eſt rectæ A D,
              <lb/>
            habebit quadr at um illius ad huius quadr at um proportionem quadruplam; </s>
            <s xml:id="echoid-s2080" xml:space="preserve">(quòd quadrata habeant du-
              <lb/>
              <note position="left" xlink:label="note-0050-06" xlink:href="note-0050-06a" xml:space="preserve">20. ſexti.</note>
              <note position="left" xlink:label="note-0050-07" xlink:href="note-0050-07a" xml:space="preserve">40</note>
            plicatam proportionem laterum) quemadmodum & </s>
            <s xml:id="echoid-s2081" xml:space="preserve">recta E B, rectæ E A, quadrupla eſt. </s>
            <s xml:id="echoid-s2082" xml:space="preserve">Rurſus quia
              <lb/>
              <note position="left" xlink:label="note-0050-08" xlink:href="note-0050-08a" xml:space="preserve">20. ſexti.</note>
            recta C G, rectæ A D, tripla est, erit quadratum illius noncuplum quadrati huius, ſicut & </s>
            <s xml:id="echoid-s2083" xml:space="preserve">recta E C,
              <lb/>
            noncupla eſt rectæ E A. </s>
            <s xml:id="echoid-s2084" xml:space="preserve">Eodem modo habebit quadratum ex H I, ad quadratum ex A D, eandem pro-
              <lb/>
            portionem, quam recta E H, ad E A, nempe ſedecuplam, & </s>
            <s xml:id="echoid-s2085" xml:space="preserve">ſic deinceps. </s>
            <s xml:id="echoid-s2086" xml:space="preserve">Quare vt conſtat ex propoſ. </s>
            <s xml:id="echoid-s2087" xml:space="preserve">20.
              <lb/>
            </s>
            <s xml:id="echoid-s2088" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2089" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2090" xml:space="preserve">Apollonij, parabola per puncta F, G, I, tranſibit. </s>
            <s xml:id="echoid-s2091" xml:space="preserve">Nam ſi per aliud punctum, vt per Q, tranſi-
              <lb/>
            re dicatur, erit ex dicta propoſ. </s>
            <s xml:id="echoid-s2092" xml:space="preserve">20. </s>
            <s xml:id="echoid-s2093" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2094" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2095" xml:space="preserve">Apollonij, quadratum ex B Q, ad quadratum ex A D, vt recta
              <lb/>
            E B, ad rectam E A, hoc eſt, vtquadratum ex B F, ad quadratum ex A D. </s>
            <s xml:id="echoid-s2096" xml:space="preserve">ſunt ergo æqualia quadrata
              <lb/>
            B Q, & </s>
            <s xml:id="echoid-s2097" xml:space="preserve">B F, & </s>
            <s xml:id="echoid-s2098" xml:space="preserve">ipſæ rectæ æquales, pars & </s>
            <s xml:id="echoid-s2099" xml:space="preserve">totum, quod est abſurdum. </s>
            <s xml:id="echoid-s2100" xml:space="preserve">Nonigitur parabola per Q,
              <lb/>
            ſed per F, deſcribenda erit, & </s>
            <s xml:id="echoid-s2101" xml:space="preserve">ſic de cæteris.</s>
            <s xml:id="echoid-s2102" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2103" xml:space="preserve">QVOD ſi quando puncta nimium inter ſe diſtare videantur, qualia ſunt G, & </s>
            <s xml:id="echoid-s2104" xml:space="preserve">I, accipiemus in dia-
              <lb/>
              <note position="left" xlink:label="note-0050-09" xlink:href="note-0050-09a" xml:space="preserve">50</note>
            metro E H, inter C, H, puncta, punctum aliquod, quod terminet particulas diametri, quas quaterna-
              <lb/>
            rius numeret, vt 8, vel 12. </s>
            <s xml:id="echoid-s2105" xml:space="preserve">vel 16. </s>
            <s xml:id="echoid-s2106" xml:space="preserve">vel 20. </s>
            <s xml:id="echoid-s2107" xml:space="preserve">&</s>
            <s xml:id="echoid-s2108" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2109" xml:space="preserve">cuiuſmodiest punctum M, terminans duodecim parti-
              <lb/>
            culas. </s>
            <s xml:id="echoid-s2110" xml:space="preserve">Deinde lineæ E M, ſumemus quartam partem, vt in dato exemplo rectam E L, continentem
              <lb/>
            tres particulas, & </s>
            <s xml:id="echoid-s2111" xml:space="preserve">ex L, perpendicularem ducemus ad E H, nempe L N, quæ parabolam ſecet in N, pun
              <lb/>
            cto. </s>
            <s xml:id="echoid-s2112" xml:space="preserve">Si enim per M, ducamus aliam perpendicularem ad E H, ex qua abſcindamus M O, duplam ipſius
              <lb/>
            L N, tranſibit parabola per punctum O; </s>
            <s xml:id="echoid-s2113" xml:space="preserve">propterca quod L M, ipſius E L, tripla eſt, & </s>
            <s xml:id="echoid-s2114" xml:space="preserve">M O, ipſius
              <lb/>
            L N, dupla, quemadmodum & </s>
            <s xml:id="echoid-s2115" xml:space="preserve">A B, ipſius E A, tripla, & </s>
            <s xml:id="echoid-s2116" xml:space="preserve">B F, ipſius A D, dupla exiſtit.</s>
            <s xml:id="echoid-s2117" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2118" xml:space="preserve">HAEC eadem ratio accommodari poteſt Parabolæ, in qua ordinatim applicatæ non ſunt perpendi-
              <lb/>
            culares ad diametrum E H, vt in conis ſcalenis contingit, cum triangulum per axem ad baſim conirectũ
              <lb/>
            non eſt, vt ex propoſ. </s>
            <s xml:id="echoid-s2119" xml:space="preserve">7. </s>
            <s xml:id="echoid-s2120" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2121" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2122" xml:space="preserve">Apoll. </s>
            <s xml:id="echoid-s2123" xml:space="preserve">liquet: </s>
            <s xml:id="echoid-s2124" xml:space="preserve">Sed tunc E H, non erit axis Parabolæ, ſed diameter. </s>
            <s xml:id="echoid-s2125" xml:space="preserve">Vn-
              <lb/>
            de per puncta A, B, C, H, ducendæ erunt lineæ inter ſe parallelæ, facientes cum diametro E H, </s>
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