Angeli, Stefano degli, Miscellaneum hyperbolicum et parabolicum : in quo praecipue agitur de centris grauitatis hyperbolae, partium eiusdem, atque nonnullorum solidorum, de quibus nunquam geometria locuta est, parabola nouiter quadratur dupliciter, ducuntur infinitarum parabolarum tangentes, assignantur maxima inscriptibilia, minimaque circumscriptibilia infinitis parabolis, conoidibus ac semifusis parabolicis aliaque geometrica noua exponuntur scitu digna

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        <div xml:id="echoid-div10" type="section" level="1" n="9">
          <head xml:id="echoid-head19" xml:space="preserve">PROPOSITIO II.</head>
          <p style="it">
            <s xml:id="echoid-s188" xml:space="preserve">Si quatuor magnitudinum ſit prima, ad ſecundam, vt tertia,
              <lb/>
            ad quartam; </s>
            <s xml:id="echoid-s189" xml:space="preserve">ſitque ablata pars primæ ad ablatam par-
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            tem ſecundæ, vt ablata pars tertiæ ad ablatam partem
              <lb/>
            quartæ et ſint partes primæ proportionales partibus ſecun-
              <lb/>
            dæ. </s>
            <s xml:id="echoid-s190" xml:space="preserve">Erit reliqua pars primæ ad reliquam partem ſecun-
              <lb/>
            dæ, vt reliqua pars tertiæ ad reliquam partem quartæ.</s>
            <s xml:id="echoid-s191" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s192" xml:space="preserve">SIT vt prima
              <lb/>
              <figure xlink:label="fig-0017-01" xlink:href="fig-0017-01a" number="6">
                <image file="0017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/PVNDER1Y/figures/0017-01"/>
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            A B, ad ſe-
              <lb/>
            cundam C D, ſic
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            tertia E F, ad
              <lb/>
            quartam G H;
              <lb/>
            </s>
            <s xml:id="echoid-s193" xml:space="preserve">ſitque k B, ad
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            L D, vt MF, ad
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            N H: </s>
            <s xml:id="echoid-s194" xml:space="preserve">pariter ſit vt Ak, ad k B, ſic E M, ad M F. </s>
            <s xml:id="echoid-s195" xml:space="preserve">
              <lb/>
            Dico etiam A K, eſſe ad C L, vt E M, ad G N. </s>
            <s xml:id="echoid-s196" xml:space="preserve">
              <lb/>
            Quoniam ex hypotheſi componendo, eſt A B, ad
              <lb/>
            B k, vt E F, ad F M; </s>
            <s xml:id="echoid-s197" xml:space="preserve">& </s>
            <s xml:id="echoid-s198" xml:space="preserve">vt k B, ad L D, ſic M F,
              <lb/>
            ad N H; </s>
            <s xml:id="echoid-s199" xml:space="preserve">ergo ex æquali, vt A B, ad L D, ſic E F,
              <lb/>
            ad N H. </s>
            <s xml:id="echoid-s200" xml:space="preserve">At pariter eſt vt A B, ad totam C D, ſic
              <lb/>
            E F, ad totam G H. </s>
            <s xml:id="echoid-s201" xml:space="preserve">Ergo & </s>
            <s xml:id="echoid-s202" xml:space="preserve">A B, erit ad reliquam
              <lb/>
            C L, vt E F, ad reliquam G N. </s>
            <s xml:id="echoid-s203" xml:space="preserve">Rurſum, quoniam
              <lb/>
            conuertendo, eſt B K, ad k A, vt F M, ad M E. </s>
            <s xml:id="echoid-s204" xml:space="preserve">
              <lb/>
            Ergo componendo, & </s>
            <s xml:id="echoid-s205" xml:space="preserve">conuertendo, erit Ak, ad A B,
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            vt EM, ad EF. </s>
            <s xml:id="echoid-s206" xml:space="preserve">Erat autem vt AB, ad CL, ſic EF, </s>
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