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

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        <div xml:id="echoid-div246" type="section" level="1" n="121">
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            <s xml:id="echoid-s4624" xml:space="preserve">
              <pb o="491" file="0213" n="224" rhead="GEOMET. VARIA."/>
            duarum E G, E F æqualem datælineæ quæ vocetur e; </s>
            <s xml:id="echoid-s4625" xml:space="preserve">& </s>
            <s xml:id="echoid-s4626" xml:space="preserve">quæ-
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            ro quanta futura ſit E G, quam appello x, ut quadrata G A,
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            G B ſimul ſumpta æquentur quadratis F A, F B.</s>
            <s xml:id="echoid-s4627" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4628" xml:space="preserve">Itaque quia A E = a, & </s>
            <s xml:id="echoid-s4629" xml:space="preserve">E G = x, erit quadratum A G =
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            aa + xx. </s>
            <s xml:id="echoid-s4630" xml:space="preserve">Et quia G D = c - x, & </s>
            <s xml:id="echoid-s4631" xml:space="preserve">D B = b, erit quadra-
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            tum G B = bb + cc - 2cx + xx, unde quadrata A G,
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            G B ſimul ſumpta fient = aa + bb + cc - 2cx + 2xx,
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            qui dicantur termini priores; </s>
            <s xml:id="echoid-s4632" xml:space="preserve">idque ſimiliter in quovis alio pro-
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            blemate intelligendum, ubi maximum aut minimum inquiri-
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            tur. </s>
            <s xml:id="echoid-s4633" xml:space="preserve">Rurfus autem quia E F = x + e, ſi ubique in ſumma
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            quadratorum inventa ſubſtituam x + e pro x, & </s>
            <s xml:id="echoid-s4634" xml:space="preserve">quadratum ab
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            x + e pro xx, adque ita deinceps ſi altior poteſtas ipſius x repc-
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            riatur, certum eſt exorituram ſummam quadratorum F A, F B;
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            </s>
            <s xml:id="echoid-s4635" xml:space="preserve">quæ quidem erit aa + bb + cc - 2cx - 2ce + 2xx + 4ex + 2ee,
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            æquanda ſummæ quadratorum A G, G B; </s>
            <s xml:id="echoid-s4636" xml:space="preserve">dicantur autem hi ter-
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            mini poſteriores.</s>
            <s xml:id="echoid-s4637" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4638" xml:space="preserve">Itaque erit aa + bb + cc - 2cx + 2xx = aa + bb + cc - 2cx
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            - 2ce + 2xx + 4ex + 2ee. </s>
            <s xml:id="echoid-s4639" xml:space="preserve">Ex qua æquatione prodibit valor E G
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            ſive x, quando G F ſive e certæ magnitudinis lineam
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            refert.</s>
            <s xml:id="echoid-s4640" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4641" xml:space="preserve">Ponendo autem e infinitè parvam, apparebit ex eadem æ-
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            quatione quanta futura ſit E G, cum ipſi E F æqualis eſt, ad-
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            eoque habebitur determinatio quæſita puncti C, unde du-
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            ctæ C A, C B faciant ſummam quadratorum minimam;
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            </s>
            <s xml:id="echoid-s4642" xml:space="preserve">nempe ſublatis primùm, ſi quæ ſunt, fractionibus, (quæ
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            in hoc exemplo nullæ ſunt) delentur termini qui utrin-
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            que iidem habentur, quales ſunt neceſſariò omnes quibus
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            litera e admixta non eſt; </s>
            <s xml:id="echoid-s4643" xml:space="preserve">idque facile eſt intelligere, cum
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            dixerimus poſteriores terminos ex prioribus deſcribi, po-
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            nendo x + e vel poteſtatem ejus, quoties invenitur x vel po-
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            teſtas ejus aliqua in prioribus. </s>
            <s xml:id="echoid-s4644" xml:space="preserve">Deinde omnes termini per
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            e dividuntur, quibuſque poſt eam diviſionem adhuc unum
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            e aut plura ineſſe inveniuntur, 11 delentur, quippe cum
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            quantitates infinitè parvas contineant reſpectu cæterorum
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            terminorum quibus nullum amplius ineſt e. </s>
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        <div xml:id="echoid-div249" type="section" level="1" n="122">
          <head xml:id="echoid-head167" style="it" xml:space="preserve">Tom. II. Qqq</head>
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            <s xml:id="echoid-s4645" xml:space="preserve">Ex quibus de-
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            nique ſolis invenitur quantitas x quæſita in caſu determina-
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            </s>
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