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

Table of Notes

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          <head xml:id="echoid-head316" xml:space="preserve">THEOR. XXXIX. PROP. LXIV.</head>
          <p>
            <s xml:id="echoid-s7529" xml:space="preserve">Portiones eiuſdem coni-ſectionis, vel circuli, aut etiam an-
              <lb/>
            guli rectilinei, quarum intercepta diametrorum ſegmenta in
              <lb/>
            Parabola ſint æqualia, vel in Hyperbola, aut in Ellipſi, vel
              <lb/>
            circulo, ad proprias ſemi- diametros eandem ſimul habeant ra-
              <lb/>
            tionem, vel in angulo pertingant ad eandem inſcriptam con-
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            centricam Hyperbolen, habent baſes altitudinibus reciprocè
              <lb/>
            proportionales.</s>
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            <s xml:id="echoid-s7531" xml:space="preserve">NAm, quo ad primùm, reiterata inſpectione figurarum tertij Schemati-
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            ſmi pro propoſitione 40. </s>
            <s xml:id="echoid-s7532" xml:space="preserve">huius; </s>
            <s xml:id="echoid-s7533" xml:space="preserve">ibi in portionibus A B C, H E I, tùm
              <lb/>
            quandò, in Parabola, diametri B F, E G ſint æquales; </s>
            <s xml:id="echoid-s7534" xml:space="preserve">tùm quandò, in
              <lb/>
              <note position="left" xlink:label="note-0270-01" xlink:href="note-0270-01a" xml:space="preserve">Schema-
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              tiſmus 3.</note>
            reliquis ſectionibus, ſit ſemi- diameter D B ad B F diametrum portionis A
              <lb/>
            B C, vt ſemi- diameter D E, ad E G diametrum portionis H E I, demon-
              <lb/>
            ſtratum ſuit, propè finem, baſim H I portionis H E I, ad baſim A C portio-
              <lb/>
            nis A B C, eſſe reciprocè, vt altitudo portionis A B C ad altitudinem por-
              <lb/>
            tionis H E I. </s>
            <s xml:id="echoid-s7535" xml:space="preserve">Quod tanquam Coroll. </s>
            <s xml:id="echoid-s7536" xml:space="preserve">Prop. </s>
            <s xml:id="echoid-s7537" xml:space="preserve">40. </s>
            <s xml:id="echoid-s7538" xml:space="preserve">huius elici poterat. </s>
            <s xml:id="echoid-s7539" xml:space="preserve">At cum
              <lb/>
            ibi tantùm loquatur de portionibus Ellipticis, quæ ſint ſemi- Ellipſi mino-
              <lb/>
            res, hoc idem verificari etiam de portionibus ſemi - Ellipſi maioribus, vel
              <lb/>
            etiam de ijſdem ſemi-Ellipſibus, ita demonſtrabitur.</s>
            <s xml:id="echoid-s7540" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s7541" xml:space="preserve">Sint duæ portiones A B C, D E F de ea-
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                <image file="0270-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0270-01"/>
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            dem Ellipſi, cuius centrum O; </s>
            <s xml:id="echoid-s7542" xml:space="preserve">vtraque ve-
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            rò ſit ſemi- Ellipſi maior, quarum diametri
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            G B, H E ad proprias ſemi - diametros B
              <lb/>
            O, E O ſint in eadem ratione. </s>
            <s xml:id="echoid-s7543" xml:space="preserve">Dico, baſim
              <lb/>
            A C vnius, ad D F baſim alterius, eſſe vt
              <lb/>
            huius altitudo E M, ad illius altitudinem
              <lb/>
            B N.</s>
            <s xml:id="echoid-s7544" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7545" xml:space="preserve">Productis enim diametris B G, E H vſq;
              <lb/>
            </s>
            <s xml:id="echoid-s7546" xml:space="preserve">ad Ellipſis peripheriam in punctis I, L, è
              <lb/>
            quibus ductis I P, L R, baſibus A C, D F
              <lb/>
            perpendicularibus, hæ erunt altitudines
              <lb/>
            portionum A I C, D L F, & </s>
            <s xml:id="echoid-s7547" xml:space="preserve">reliquarum
              <lb/>
            portionum altitudinibus, B N, E M æqui-
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            diſtabunt.</s>
            <s xml:id="echoid-s7548" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s7549" xml:space="preserve">Et cum, ex hypotheſi, ſit G B ad B O, vt H E ad E O, ſumptis conſe-
              <lb/>
            quentium duplis, conuertendo, & </s>
            <s xml:id="echoid-s7550" xml:space="preserve">per conuerſionem rationis B I ad I G,
              <lb/>
            erit vt E L ad L H; </s>
            <s xml:id="echoid-s7551" xml:space="preserve">& </s>
            <s xml:id="echoid-s7552" xml:space="preserve">ſumptis antecedentium ſubduplis, O I ad I G, vt O
              <lb/>
            L ad L H: </s>
            <s xml:id="echoid-s7553" xml:space="preserve">quare, per ſuperiùs oſtenſa, in portionibus A I C, D L F, ſemi-
              <lb/>
            Ellipſi minoribus, erit baſis A C ad D F, vt altitudo L R ad altitudinem
              <lb/>
            I P, fed L R ad I P eſt, vt E M ad B N, vt mox demonſtrabitur, ergo A
              <lb/>
            C ad D F erit quoque, vt E M ad B N.</s>
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          <p>
            <s xml:id="echoid-s7555" xml:space="preserve">Quod autem ſit L R ad I P, vt E M ad B N. </s>
            <s xml:id="echoid-s7556" xml:space="preserve">Cum demonſtratum </s>
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