Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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            <s xml:id="echoid-s2120" xml:space="preserve">
              <pb o="392" file="0106" n="113" rhead="CHRISTIANI HUGENII"/>
            us ducenda eſt: </s>
            <s xml:id="echoid-s2121" xml:space="preserve">A D vero ſic, ut ſubtenſa C D æqualis ſit
              <lb/>
            abſciſſæ E F. </s>
            <s xml:id="echoid-s2122" xml:space="preserve">Etenim his poſitis dico cubum A E ejus qui
              <lb/>
            ex A B duplum exiſtere.</s>
            <s xml:id="echoid-s2123" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2124" xml:space="preserve">Producatur enim C A, & </s>
            <s xml:id="echoid-s2125" xml:space="preserve">ſit ipſi A E æqualis A G.
              <lb/>
            </s>
            <s xml:id="echoid-s2126" xml:space="preserve">Propter triangulos ſimiles igitur eſt E C ad C D, hoc eſt,
              <lb/>
            E F ut E A ad A F, hoc eſt, ut G A ad A B. </s>
            <s xml:id="echoid-s2127" xml:space="preserve">Et com-
              <lb/>
            ponendo C F ad F E ut G B ad B A ſive A F. </s>
            <s xml:id="echoid-s2128" xml:space="preserve">Et permu-
              <lb/>
            tando C F ad G B ut E F ad F A. </s>
            <s xml:id="echoid-s2129" xml:space="preserve">Quare ut C F qua-
              <lb/>
            dratum ad quadr. </s>
            <s xml:id="echoid-s2130" xml:space="preserve">G B, ita quadr. </s>
            <s xml:id="echoid-s2131" xml:space="preserve">E F ad quadr. </s>
            <s xml:id="echoid-s2132" xml:space="preserve">F A. </s>
            <s xml:id="echoid-s2133" xml:space="preserve">
              <lb/>
            Et componendo ut quadr. </s>
            <s xml:id="echoid-s2134" xml:space="preserve">C F & </s>
            <s xml:id="echoid-s2135" xml:space="preserve">G B ad quadr. </s>
            <s xml:id="echoid-s2136" xml:space="preserve">G B, ita
              <lb/>
            quadr. </s>
            <s xml:id="echoid-s2137" xml:space="preserve">E F & </s>
            <s xml:id="echoid-s2138" xml:space="preserve">F A ſimul, hoc eſt, quadr. </s>
            <s xml:id="echoid-s2139" xml:space="preserve">E A ad quadr. </s>
            <s xml:id="echoid-s2140" xml:space="preserve">
              <lb/>
            A F. </s>
            <s xml:id="echoid-s2141" xml:space="preserve">Quadr. </s>
            <s xml:id="echoid-s2142" xml:space="preserve">autem C F & </s>
            <s xml:id="echoid-s2143" xml:space="preserve">G B ſimul æquantur rectangu-
              <lb/>
            lo G C A cum quadr. </s>
            <s xml:id="echoid-s2144" xml:space="preserve">A G, quod ſic oſtenditur. </s>
            <s xml:id="echoid-s2145" xml:space="preserve">Quadra-
              <lb/>
            tum enim G B æquale eſt rectangulo C G A & </s>
            <s xml:id="echoid-s2146" xml:space="preserve">quadrato
              <lb/>
            A B ſeu A F . </s>
            <s xml:id="echoid-s2147" xml:space="preserve">Quare addito utrimque quadrato F
              <note symbol="*" position="left" xlink:label="note-0106-01" xlink:href="note-0106-01a" xml:space="preserve">6.2. Elem.</note>
            erunt quadrata G B, F C ſimul æqualia rectangulo C G A
              <lb/>
            & </s>
            <s xml:id="echoid-s2148" xml:space="preserve">quadrato A C. </s>
            <s xml:id="echoid-s2149" xml:space="preserve">Rectangulum autem C G A cum quadra-
              <lb/>
            to A C æquatur rectangulo G C A cum quadrato G A. </s>
            <s xml:id="echoid-s2150" xml:space="preserve">Ita-
              <lb/>
            que & </s>
            <s xml:id="echoid-s2151" xml:space="preserve">quadrata C F, G B ſimul æqualia ſunt rectangulo
              <lb/>
            G C A cum quadrato A G, ſicut diximus. </s>
            <s xml:id="echoid-s2152" xml:space="preserve">Sicut igitur re-
              <lb/>
            ctangulum G C A cum quadrato A G ad quadr. </s>
            <s xml:id="echoid-s2153" xml:space="preserve">G B, ita
              <lb/>
            eſt quadr. </s>
            <s xml:id="echoid-s2154" xml:space="preserve">E A ad quadr. </s>
            <s xml:id="echoid-s2155" xml:space="preserve">A F, hoc eſt, ita quadratum
              <lb/>
            G A ad quadr. </s>
            <s xml:id="echoid-s2156" xml:space="preserve">A B. </s>
            <s xml:id="echoid-s2157" xml:space="preserve">Et permutando, ut rectangulum
              <lb/>
            G C A cum quadrato G A ad quadratum G A ita quadr.
              <lb/>
            </s>
            <s xml:id="echoid-s2158" xml:space="preserve">G B ad quadr. </s>
            <s xml:id="echoid-s2159" xml:space="preserve">A B. </s>
            <s xml:id="echoid-s2160" xml:space="preserve">Dividendo igitur, erit ut rectang. </s>
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              <lb/>
            G C A ad quadr. </s>
            <s xml:id="echoid-s2162" xml:space="preserve">G A, ita quadr. </s>
            <s xml:id="echoid-s2163" xml:space="preserve">G B dempto quadrato
              <lb/>
            A B, hoc eſt, rectangulum C G A ad quadr. </s>
            <s xml:id="echoid-s2164" xml:space="preserve">A B. </s>
            <s xml:id="echoid-s2165" xml:space="preserve">Et per-
              <lb/>
            mutando rurſus, ut rectang. </s>
            <s xml:id="echoid-s2166" xml:space="preserve">G C A ad rectang. </s>
            <s xml:id="echoid-s2167" xml:space="preserve">C G A,
              <lb/>
            hoc eſt, ut C A ad A G ita quadratum G A ad quadr. </s>
            <s xml:id="echoid-s2168" xml:space="preserve">
              <lb/>
            A B. </s>
            <s xml:id="echoid-s2169" xml:space="preserve">Quamobrem quod fit ex quadrato G A in ipſam
              <lb/>
            G A, hoc eſt, cubus G A æquabitur ei quod fit ex quadra-
              <lb/>
            to A B in A C, hoc eſt, duplo cubo ex A B. </s>
            <s xml:id="echoid-s2170" xml:space="preserve">Quod erat
              <lb/>
            demonſtrandum.</s>
            <s xml:id="echoid-s2171" xml:space="preserve"/>
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