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

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              <pb o="431" file="0149" n="158" rhead="ET HYPERBOLÆ QUADRATURA."/>
            æqualibus quantitatibus addendo, ſubtrahendo, multiplican-
              <lb/>
            do, dividendo, &</s>
            <s xml:id="echoid-s3193" xml:space="preserve">c: </s>
            <s xml:id="echoid-s3194" xml:space="preserve">ſicut ſuperius dictum eſt, ſemper ma-
              <lb/>
            net poteſtas ipſius a in ultimo producto ex termino a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b
              <lb/>
            altior poteſtate ulla ipſius a in ultimo producto termini
              <lb/>
            ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, quoniam illos cum æqualibus quantitatibus ad-
              <lb/>
            dendo, ſubſtrahendo, &</s>
            <s xml:id="echoid-s3195" xml:space="preserve">c, ſemper manent in factis eædem
              <lb/>
            poteſtates, & </s>
            <s xml:id="echoid-s3196" xml:space="preserve">illos in ſe eodem modo multiplicando ſemper
              <lb/>
            magis elevatur altior poteſtas in termino a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b quam ele-
              <lb/>
            vatur depreſſior poteſtas in termino ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, & </s>
            <s xml:id="echoid-s3197" xml:space="preserve">ex illis
              <lb/>
            eaſdem radices extrahendo, ubi a fuerat elevatior in poteſta-
              <lb/>
            te erit etiam elevatior in radice: </s>
            <s xml:id="echoid-s3198" xml:space="preserve">& </s>
            <s xml:id="echoid-s3199" xml:space="preserve">quoniam eadem reperitur
              <lb/>
            altiſſima poteſtas ipſius a in termino ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            quæ reperitur
              <lb/>
            in termino 2b
              <emph style="super">2</emph>
            a, demonſtratur ut antè altiſſimam poteſta-
              <lb/>
            tem ipſius a in ultimo producto ex termino ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            eandem
              <lb/>
            eſſe cum altiſſima poteſtate ipſius a in ultimo producto ex
              <lb/>
            termino 2b
              <emph style="super">2</emph>
            a: </s>
            <s xml:id="echoid-s3200" xml:space="preserve">in ultimo igitur producto termini a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b
              <lb/>
            reperitur altior poteſtas ipſius a quam in ultimo producto
              <lb/>
            termini ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, & </s>
            <s xml:id="echoid-s3201" xml:space="preserve">in ultimo producto termini ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
              <lb/>
            altiſſima poteſtas ipſius a eadem eſt cum altiſſima poteſtate
              <lb/>
            ipſius a in ultimo producto termini 2b
              <emph style="super">2</emph>
            a; </s>
            <s xml:id="echoid-s3202" xml:space="preserve">& </s>
            <s xml:id="echoid-s3203" xml:space="preserve">igitur ultima
              <lb/>
            producta ex terminis a
              <emph style="super">3</emph>
            + a
              <emph style="super">2</emph>
            b, ab
              <emph style="super">2</emph>
            + b
              <emph style="super">3</emph>
            , eodem modo inter
              <lb/>
            ſe addita, ſubducta, multiplicata, diviſa, &</s>
            <s xml:id="echoid-s3204" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3205" xml:space="preserve">ſemper effi-
              <lb/>
            cient quantitatem, in qua reperitur altior poteſtas ipſius a
              <lb/>
            quam ulla quæ reperiri poteſt in quantitate facta ex eadem
              <lb/>
            prorſus additione, ſubductione, multiplicatione, diviſione,
              <lb/>
            &</s>
            <s xml:id="echoid-s3206" xml:space="preserve">c, productorum à terminis ba
              <emph style="super">2</emph>
            + b
              <emph style="super">2</emph>
            a, 2b
              <emph style="super">2</emph>
            a; </s>
            <s xml:id="echoid-s3207" xml:space="preserve">quoniam al-
              <lb/>
            tior poteſtas cum altera poteſtate ſemper altiorem facit po-
              <lb/>
            teſtatem quam depreſſior poteſtas cum eadem altera poteſta-
              <lb/>
            te; </s>
            <s xml:id="echoid-s3208" xml:space="preserve">& </s>
            <s xml:id="echoid-s3209" xml:space="preserve">igitur iſtæ duæ quantitates non poſſunt eſſe indefini-
              <lb/>
            tè æquales, cum reperiatur altior poteſtas ipſius a in una
              <lb/>
            quam in altera: </s>
            <s xml:id="echoid-s3210" xml:space="preserve">atque hinc evidens eſt quod ſector circuli,
              <lb/>
            ellipſeos vel hyperbolæ A B I P non poſſit componi analy-
              <lb/>
            ticè à triangulo A B P & </s>
            <s xml:id="echoid-s3211" xml:space="preserve">trapezio A B F P, quod demon-
              <lb/>
            ſtrandum erat.</s>
            <s xml:id="echoid-s3212" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3213" xml:space="preserve">Ut autem evidentius fiat propoſitum, aliam demonſtratio-
              <lb/>
            nem treviorem faciliorem & </s>
            <s xml:id="echoid-s3214" xml:space="preserve">ex alio medio petitam hic </s>
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