Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

Table of Notes

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            ſed quadratum MA minus eſt quadrato HM, ergo quadratum A I
              <note symbol="a" position="left" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve">87. h.</note>
            erit quadrato HI, ſiue perpendicularis intercepta A I, maior intercepto mi-
              <lb/>
            noris axis ſegmento IH. </s>
            <s xml:id="echoid-s4572" xml:space="preserve">Quod tandem demonſtrare oportebat.</s>
            <s xml:id="echoid-s4573" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s4574" xml:space="preserve">ALITER abſque ope propoſitionis 87. </s>
            <s xml:id="echoid-s4575" xml:space="preserve">premiſso
              <lb/>
            tantum ſequenti lemmate pro Ellipſi.</s>
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        <div xml:id="echoid-div448" type="section" level="1" n="186">
          <head xml:id="echoid-head191" xml:space="preserve">LEMMA XIII. PROP. XIC.</head>
          <p>
            <s xml:id="echoid-s4577" xml:space="preserve">Si ſuerit, in vtraque figura, rectangulum ſub extremis AB, BD
              <lb/>
            æquale quadrato mediæ BC, dico, in prima ſigura, ſi à tertia BD
              <lb/>
            dematur aliqua pars BE, rectangulum ſub AE, ED, minus eſſe
              <lb/>
            quadrato mediæ EC.</s>
            <s xml:id="echoid-s4578" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4579" xml:space="preserve">Cum ſit enim, vt totum AB ad totum BC, ita ablatum BC ad ablatũ BD,
              <lb/>
            erit reliquum AC ad reliquum CD, vt totum AB ad totum BC.</s>
            <s xml:id="echoid-s4580" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4581" xml:space="preserve">Et cum ſit CE minor C B, habebit
              <lb/>
              <figure xlink:label="fig-0160-01" xlink:href="fig-0160-01a" number="126">
                <image file="0160-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0160-01"/>
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            AC ad CE maiorem rationem quàm
              <lb/>
            AC ad CB, & </s>
            <s xml:id="echoid-s4582" xml:space="preserve">componendo AE ad
              <lb/>
            EC maiorem quàm AB ad BC, vel
              <lb/>
            quàm AC ad CD. </s>
            <s xml:id="echoid-s4583" xml:space="preserve">Siergo totum AE
              <lb/>
            ad totum EC maioré habet rationem
              <lb/>
            quàm ablatum AC ad ablatum CD,
              <lb/>
            habebit reliquum CE ad reliquũ ED
              <lb/>
            maiorem rationem, quàm totum AE
              <lb/>
              <note symbol="b" position="left" xlink:label="note-0160-02" xlink:href="note-0160-02a" xml:space="preserve">16. 7.
                <lb/>
              Pappi.</note>
            ad totum EC, vel AE ad EC minorem
              <lb/>
            habebit rationem quàm CE ad ED;
              <lb/>
            </s>
            <s xml:id="echoid-s4584" xml:space="preserve">ergo rectangulum ſub extremis A E,
              <lb/>
            ED minus erit quadrato mediæ EC.</s>
            <s xml:id="echoid-s4585" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4586" xml:space="preserve">SI verò, ijſdem poſitis, in ſecunda ſigura, tertiæ proportionali BD recta
              <lb/>
            quædam BE adijciatur; </s>
            <s xml:id="echoid-s4587" xml:space="preserve">dico rectangulum ſub AE, ED maius eſſe qua-
              <lb/>
            drato EC; </s>
            <s xml:id="echoid-s4588" xml:space="preserve">quod licet in 9. </s>
            <s xml:id="echoid-s4589" xml:space="preserve">prop. </s>
            <s xml:id="echoid-s4590" xml:space="preserve">huius iam ſit oſtenſum, hic idem aliter nulla
              <lb/>
            facta conſtructione demonſtrabimus.</s>
            <s xml:id="echoid-s4591" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4592" xml:space="preserve">Quoniam enim CE maior eſt CB, habebit AC ad CE minorem rationem
              <lb/>
            quàm AC ad CB, & </s>
            <s xml:id="echoid-s4593" xml:space="preserve">componendo, tota AE ad totam EC, minorem quàm
              <lb/>
              <note symbol="c" position="left" xlink:label="note-0160-03" xlink:href="note-0160-03a" xml:space="preserve">ibidem.</note>
            ablata AB ad ablatam BC, vel quàm AC ad CD, ergo reliqua CE ad re-
              <lb/>
            liquam ED, minorem quoque habebit rationem quàm tota AE ad EC,
              <lb/>
            hoc eſt AE ad EC maiorem quàm EC ad ED, ergo rectangulum ſub AE,
              <lb/>
            ED maius quadrato mediæ EC. </s>
            <s xml:id="echoid-s4594" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s4595" xml:space="preserve">c.</s>
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            <s xml:id="echoid-s4597" xml:space="preserve">IAM, vt ad expeditiorem demonſtrationem præcedentis propoſitionis ac-
              <lb/>
            cedamus, ſuper eiſdem delineationibus, repetitis ijs omnibus, quæ ibi
              <lb/>
            (vſque ad ea verba excluſiuè _Ducta enim ex B recta BG, &</s>
            <s xml:id="echoid-s4598" xml:space="preserve">c.)</s>
            <s xml:id="echoid-s4599" xml:space="preserve">_ exponuntur, ac
              <lb/>
            demonſtrantur, ſic vlteriùs proſequemur. </s>
            <s xml:id="echoid-s4600" xml:space="preserve">Cum enim in ſingulis figuris triã-
              <lb/>
            gula DAE, LAI ſint rectangula ad A, ex quo baſibus ductæ ſunt perpendi-
              <lb/>
            culares AF, AR; </s>
            <s xml:id="echoid-s4601" xml:space="preserve">erit in triangulo DAE rectangulum EDF æquale </s>
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