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

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            <s xml:id="echoid-s4819" xml:space="preserve">
              <pb o="501" file="0223" n="234" rhead="GEOMET. VARIA."/>
            atque A F, F B, hoc eſt x & </s>
            <s xml:id="echoid-s4820" xml:space="preserve">y. </s>
            <s xml:id="echoid-s4821" xml:space="preserve">Nempe ſi in æquatione
              <lb/>
            propoſita pro x ſubſtituatur ubique x + e, & </s>
            <s xml:id="echoid-s4822" xml:space="preserve">pro y, ubique
              <lb/>
            y + {ey/z}, debebit æquatio hinc formata terminos omnes ha-
              <lb/>
            bere æquales nihilo; </s>
            <s xml:id="echoid-s4823" xml:space="preserve">hoc eſt
              <lb/>
            x
              <emph style="super">3</emph>
            + [3exx] + 3eex + e
              <emph style="super">3</emph>
            , + y
              <emph style="super">3</emph>
            + [{3ey
              <emph style="super">3</emph>
            /z}] + {3eey
              <emph style="super">3</emph>
            /zz} + {e
              <emph style="super">3</emph>
            y
              <emph style="super">3</emph>
            /z
              <emph style="super">3</emph>
            } = 0.
              <lb/>
            </s>
            <s xml:id="echoid-s4824" xml:space="preserve">- axy - [aey] - [{aeyx/z}] - {aeey/z}</s>
          </p>
          <p>
            <s xml:id="echoid-s4825" xml:space="preserve">In hac autem æquatione conſtat neceſſario terminos prio-
              <lb/>
            ris æquationis, ex qua formata eſt, contineri debere, nem-
              <lb/>
            pe x
              <emph style="super">3</emph>
            + y
              <emph style="super">3</emph>
            - axy: </s>
            <s xml:id="echoid-s4826" xml:space="preserve">qui cum ſint æquales nihilo ex proprie-
              <lb/>
            tate curvæ, idcirco his in æquatione deletis, neceſſe eſt etiam
              <lb/>
            reliquos nihilo æquari, in quibus ſingulis manifeſtum quoque
              <lb/>
            eſt vel unum e vel plura reperiri, ideoque omnes per e divi-
              <lb/>
            di poſſe. </s>
            <s xml:id="echoid-s4827" xml:space="preserve">Qui autem poſt hanc diviſionem non amplius ha-
              <lb/>
            bebunt e, eos, neglectis reliquis, ſcio nihilo æquari debe-
              <lb/>
            re, quantitatemque lineæ z ſive F E oſtenſuros; </s>
            <s xml:id="echoid-s4828" xml:space="preserve">ſi nempe
              <lb/>
            B E jam tanquam tangens conſideretur, ideoque F G, ſeu
              <lb/>
            e, infinitè parva. </s>
            <s xml:id="echoid-s4829" xml:space="preserve">Nam termini in quibus adhuc e ſupereſt,
              <lb/>
            etiam quantitates infinite parvas ſive omnino evaneſcentes
              <lb/>
            continebunt. </s>
            <s xml:id="echoid-s4830" xml:space="preserve">Et his quidem hactenus Fermatianæ regulæ
              <lb/>
            origo ac ratio declaratur: </s>
            <s xml:id="echoid-s4831" xml:space="preserve">nunc porro oſtendemus quomodo
              <lb/>
            eadem ad tantam brevitatem perducta ſit. </s>
            <s xml:id="echoid-s4832" xml:space="preserve">Video itaque ex
              <lb/>
            æquatione totâ noviſſimâ, tantum eos terminos ſeribi neceſ-
              <lb/>
            ſe eſſe quibus ineſt e ſimplex, velut hic 3exx + {3ey
              <emph style="super">3</emph>
            /z} - aey -
              <lb/>
            {aeyx/z} = 0. </s>
            <s xml:id="echoid-s4833" xml:space="preserve">Qui termini quomodo facili negotio ex datis æqua-
              <lb/>
            tionis terminis x
              <emph style="super">3</emph>
            + y
              <emph style="super">3</emph>
            - axy = 0, deſcribi poſſint, dein-
              <lb/>
            ceps explicandum. </s>
            <s xml:id="echoid-s4834" xml:space="preserve">Et primò quidem apparet 3exx +
              <lb/>
            {3ey
              <emph style="super">3</emph>
            /z} nihil aliud eſſe quam ſecundos terminos cuborum </s>
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