Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div484" type="section" level="1" n="177">
          <p style="it">
            <s xml:id="echoid-s7834" xml:space="preserve">
              <pb o="184" file="214" n="214" rhead="GEOMETR. PRACT."/>
            KAO; </s>
            <s xml:id="echoid-s7835" xml:space="preserve">ac proinde multo mai{us}, quam dimidium trianguli mixti KAO, cui{us} vnum la-
              <lb/>
            t{us} eſt arc{us} A O. </s>
            <s xml:id="echoid-s7836" xml:space="preserve">Eadem ratione erit K O X, mai{us}, quam dimidium trianguli mixti
              <lb/>
            KOB, cui{us} vnum lat{us} eſt arc{us} O B. </s>
            <s xml:id="echoid-s7837" xml:space="preserve">Auferendo ergo triangulum KVX, ex figura mi-
              <lb/>
            ſtilinea K A B, in qua vnum lat{us} eſt arc{us} A O B, ablatum erit pl{us} quam dimidium.
              <lb/>
            </s>
            <s xml:id="echoid-s7838" xml:space="preserve">Subtractis igitur quatuor eiuſmodi triangulis K V X, L Y a, M b c, I d e, ablatum erit
              <lb/>
            plus, quam dimidium ex quatuor reſiduis extra circulum, & </s>
            <s xml:id="echoid-s7839" xml:space="preserve">ſic deinceps. </s>
            <s xml:id="echoid-s7840" xml:space="preserve">Ponantur
              <lb/>
            igituriam octo triãgula mixta reſidua, quorũ baſes ſunt arcus AO, OB, BP, &</s>
            <s xml:id="echoid-s7841" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7842" xml:space="preserve">
              <lb/>
            minora magnitudine z. </s>
            <s xml:id="echoid-s7843" xml:space="preserve">Cum ergo circulus cum z, æqualis poſitus ſit triangulo
              <lb/>
            F G H, erit circulus cum illis octo reſiduis, hoc eſt, figura Octogona V X Y,
              <lb/>
            a b c d e V, minor eodem triangulo F G H. </s>
            <s xml:id="echoid-s7844" xml:space="preserve">quod eſt ab ſurdum, cum maius fit: </s>
            <s xml:id="echoid-s7845" xml:space="preserve">
              <lb/>
            quippe cum perpẽdicularis EO, æqualis ſit lateri F G, & </s>
            <s xml:id="echoid-s7846" xml:space="preserve">ambitus Octogini ma-
              <lb/>
            ior circumferentia circuli, hoc eſt, recta GH. </s>
            <s xml:id="echoid-s7847" xml:space="preserve">Hinc enim fit, triangulum rectan-
              <lb/>
            gulum, cuius latus F G, æquale perpendiculari EO, & </s>
            <s xml:id="echoid-s7848" xml:space="preserve">alterum latus æquale am-
              <lb/>
            bitui Octogoni, maius videlicet, quam GH, maius eſſe triangulo FGH. </s>
            <s xml:id="echoid-s7849" xml:space="preserve">Cum er-
              <lb/>
            go illud triangulum ſit, ex ſcholio propoſ. </s>
            <s xml:id="echoid-s7850" xml:space="preserve">41. </s>
            <s xml:id="echoid-s7851" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7852" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7853" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s7854" xml:space="preserve">æquale rectangulo ſub
              <lb/>
            FG, & </s>
            <s xml:id="echoid-s7855" xml:space="preserve">ſemiſſe ambitus Octogoni comprehenſo; </s>
            <s xml:id="echoid-s7856" xml:space="preserve">hoc autem rectangulum O-
              <lb/>
            ctogono æquale, ex propoſ. </s>
            <s xml:id="echoid-s7857" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7858" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7859" xml:space="preserve">7. </s>
            <s xml:id="echoid-s7860" xml:space="preserve">de Iſoperimetris: </s>
            <s xml:id="echoid-s7861" xml:space="preserve">erit quoq; </s>
            <s xml:id="echoid-s7862" xml:space="preserve">Octogonum
              <lb/>
            maius triangulo FGH. </s>
            <s xml:id="echoid-s7863" xml:space="preserve">Nõ ergo minus eſſe poteſt, ac proinde circulus ABCD,
              <lb/>
            minor non eſt triangulo FGH: </s>
            <s xml:id="echoid-s7864" xml:space="preserve">Sed neque maior eſt, vt demonſtrauimus. </s>
            <s xml:id="echoid-s7865" xml:space="preserve">Igi-
              <lb/>
            tur æqualis eſt, quod erat demonſtrandum.</s>
            <s xml:id="echoid-s7866" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div489" type="section" level="1" n="178">
          <head xml:id="echoid-head185" xml:space="preserve">SCHOLIVM.</head>
          <p>
            <s xml:id="echoid-s7867" xml:space="preserve">
              <emph style="sc">Iosephvs</emph>
            Scaliger, vel quia vim huius demonſtrationis non perpendit,
              <lb/>
            vel quia ſuæ circuli quadrandi rationi vidit eſſe contrariam, non eſt veritus Ar-
              <lb/>
            chimedem hoc loco falſitatis arguere: </s>
            <s xml:id="echoid-s7868" xml:space="preserve">conaturque oſtendere, non rectè ab eo
              <lb/>
            demonſtratum, circulum æqualem eſſe triangulo rectangulo, cuius vnum latus
              <lb/>
            ſemidiametro, & </s>
            <s xml:id="echoid-s7869" xml:space="preserve">alterum circumferentiæ circuli eſt æquale. </s>
            <s xml:id="echoid-s7870" xml:space="preserve">Nam, ait, ſi de-
              <lb/>
            monſtratio Archimedis bona eſt, demonſtrabitur eodem modo, circulũ æqua-
              <lb/>
            lem eſſe triangulo rectangulo, cuius vnum latus circa angulum rectum ſemidi-
              <lb/>
            ametro æquale eſt, & </s>
            <s xml:id="echoid-s7871" xml:space="preserve">alterum peripheria circuli maius. </s>
            <s xml:id="echoid-s7872" xml:space="preserve">Sit enim in triangulo
              <lb/>
            lmn, latus quidem lm, trianguli ſemidiametro circuli E A, æquale, at mn, peri-
              <lb/>
            pheria maius. </s>
            <s xml:id="echoid-s7873" xml:space="preserve">Concedit ergo Scaliger, circulum non eſſe maiorem triangulo
              <lb/>
            FGH, rectè eſſe ab Archimede demonſtratum, hoc eſt, triangulum F G H, cuius
              <lb/>
            latus GH, peripheriæ eſt æquale, non eſſe minus circulo, ac proinde neque tri-
              <lb/>
            angulum lmn, cuius latus m n, maius eſt peripheria, circulo minus eſſe. </s>
            <s xml:id="echoid-s7874" xml:space="preserve">Con-
              <lb/>
            cedititem, rectè probatum eſſe, circulũ non eſſe minorem triangulo FGH, ſi la-
              <lb/>
            tus GH, peripheriæ ſit ęquale, hoc eſt, triangulum FGH, non eſſe maius circu-
              <lb/>
            lo. </s>
            <s xml:id="echoid-s7875" xml:space="preserve">Sed negat, ex hoc ſequi, triangulum FGH, eſſe æquale circulo. </s>
            <s xml:id="echoid-s7876" xml:space="preserve">Cur? </s>
            <s xml:id="echoid-s7877" xml:space="preserve">quia,
              <lb/>
            inquit, eodem modo, ſi baſis m n, maior eſt peripheria, ſed minor circumſcripti
              <lb/>
            polygoni ambitu, (hoc enim contingere, ait, nihil prohibet) polygonum erit
              <lb/>
            quidem maius triangulo l m n, quod ambitus polygoni maior ſit recta m n, & </s>
            <s xml:id="echoid-s7878" xml:space="preserve">
              <lb/>
            ſemidiameter EA, rectæ l m, æqualis. </s>
            <s xml:id="echoid-s7879" xml:space="preserve">Sed reſectis portionibus, ſequeretur, idẽ
              <lb/>
            polygonum eſſe triangulo l m n, minus, quod eſt ineptum. </s>
            <s xml:id="echoid-s7880" xml:space="preserve">Ita ne verò mi Sca-
              <lb/>
            liger? </s>
            <s xml:id="echoid-s7881" xml:space="preserve">Non aduertis, te cum hypotheſi pugnare? </s>
            <s xml:id="echoid-s7882" xml:space="preserve">Nam poſito latere m n, ma-
              <lb/>
            iore, quam peripheria; </s>
            <s xml:id="echoid-s7883" xml:space="preserve">quando eo peruentum erit, polygonum eſſe minus tri-
              <lb/>
            angulo l m n, (ſi nimirumrelictæ portiones minores fuerint magnitudine z,) </s>
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