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

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            <s xml:id="echoid-s4903" xml:space="preserve">
              <pb o="506" file="0228" n="239" rhead="CHRIST. HUGENII"/>
            ſive etiam - 3y
              <emph style="super">3</emph>
            + ayx eſſe affirmatam, ac proinde 3y
              <emph style="super">3</emph>
            - ayx
              <lb/>
            eſſe negatam: </s>
            <s xml:id="echoid-s4904" xml:space="preserve">aut quando 3xx - ay fuerit affirmata, tunc
              <lb/>
            - 3y
              <emph style="super">3</emph>
            + ayx eſſe negatam; </s>
            <s xml:id="echoid-s4905" xml:space="preserve">ac proinde 3y
              <emph style="super">3</emph>
            - ayx eſſe affir-
              <lb/>
            matam. </s>
            <s xml:id="echoid-s4906" xml:space="preserve">Per hæc itaque apparet ex quantitatibus per regulam in-
              <lb/>
            ventis, quæ erant {3y
              <emph style="super">3</emph>
            - ayx/3xx - ay} = z judiciari poſſe ad utrum ca-
              <lb/>
            ſum conſtructio tangentis pertineat; </s>
            <s xml:id="echoid-s4907" xml:space="preserve">nempe excomperta diſſi-
              <lb/>
            militudine affectionis in diviſore & </s>
            <s xml:id="echoid-s4908" xml:space="preserve">dividendo, ſequi ad prio-
              <lb/>
            rem caſum eam pertinere, hoc eſt z, ſive F E, accipiendam
              <lb/>
            eſſe verſus A: </s>
            <s xml:id="echoid-s4909" xml:space="preserve">ex ſimilitudine vero eorum affectionis ſequi ad
              <lb/>
            contrariam partem ſumendam.</s>
            <s xml:id="echoid-s4910" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">TAB. XLV.
            <lb/>
          fig. 6.</note>
          <p>
            <s xml:id="echoid-s4911" xml:space="preserve">Poteſt autem quantitas z ſive FE per regulam inventa, non-
              <lb/>
            nunquam ad ſimpliciores terminos reduci ope æquationis datæ,
              <lb/>
            quæ naturam curvæ continet: </s>
            <s xml:id="echoid-s4912" xml:space="preserve">velut in hac curva A C, axem
              <lb/>
            habente A D, verticem A, cujuſque ea eſt proprietas ut, ſi à
              <lb/>
            puncto C in eâ ſumpta, applicetur ordinatim C D, fiat pro-
              <lb/>
            ductum ex cubo B D (eſt autem B punctum in axe extra cur-
              <lb/>
            vam datam) in quadratum D A æquale cubo quadrato DC. </s>
            <s xml:id="echoid-s4913" xml:space="preserve">Si-
              <lb/>
            ve ponendo B A = a, B D = x, D C = y, fiat æquatio cur-
              <lb/>
            væ naturam continens, iſta x
              <emph style="super">5</emph>
            - 2ax
              <emph style="super">4</emph>
            + aax
              <emph style="super">3</emph>
            - y
              <emph style="super">5</emph>
            = o. </s>
            <s xml:id="echoid-s4914" xml:space="preserve">Hîc
              <lb/>
            ponendo CG eſſe tangentem, quæ occurrat axi in G, vocan-
              <lb/>
            doque GD, z, fit ſecundum regulam z = {- 5y
              <emph style="super">5</emph>
            /5x
              <emph style="super">4</emph>
            - 8ax
              <emph style="super">3</emph>
            + 3aaxx}.
              <lb/>
            </s>
            <s xml:id="echoid-s4915" xml:space="preserve">Quia autem ex datâ æquatione eſt y
              <emph style="super">5</emph>
            = x
              <emph style="super">5</emph>
            - 2ax
              <emph style="super">4</emph>
            + aax
              <emph style="super">3</emph>
            , re-
              <lb/>
            ſtituendo pro 5y id quod ipſi æquale eſt, fiet z =
              <lb/>
            {- 5x
              <emph style="super">5</emph>
            + 10ax
              <emph style="super">4</emph>
            - 5aax
              <emph style="super">3</emph>
            /5x
              <emph style="super">4</emph>
            - 8ax
              <emph style="super">3</emph>
            + 3aaxx}; </s>
            <s xml:id="echoid-s4916" xml:space="preserve">ſive dividendo per xx, erit z =
              <lb/>
            {- 5x
              <emph style="super">3</emph>
            + 10axx - 5aax/5xx - 8ax + 3aa,}. </s>
            <s xml:id="echoid-s4917" xml:space="preserve">Et rurſus, dividendo hanc fractionem
              <lb/>
            per x - a, habebitur z = {-5xx + 5ax/5x - 3a}. </s>
            <s xml:id="echoid-s4918" xml:space="preserve">Quod ſignificat
              <lb/>
            faciendum ut ſicut B D quinquies ſumpta minus B A ter, ſive
              <lb/>
            ut B A bis unà cum A D quinquies ad AD quinquies, ita BD
              <lb/>
            ad D G; </s>
            <s xml:id="echoid-s4919" xml:space="preserve">atque ita G C tacturam in C curvam A C.</s>
            <s xml:id="echoid-s4920" xml:space="preserve"/>
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