Blancanus, Josephus, Sphaera mvndi, sev cosmographia demonstratiua , ac facile methodo tradita : in qua totius Mundi fabrica, vna cum nouis, Tychonis, Kepleri, Galilaei, aliorumq' ; Astronomorum adinuentis continentur ; Accessere I. Breuis introductio ad geographiam. II. Apparatus ad mathematicarum studium. III. Echometria, idest Geometrica tractatio de Echo. IV. Nouum instrumentum ad Horologia

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    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div57" type="section" level="1" n="39">
          <pb o="29" file="0045" n="45" rhead="Liber Secundus."/>
          <p>
            <s xml:id="echoid-s2877" xml:space="preserve">Quod autem ſphæra ſit omnium figurarum, tam planarum, quam ſolidarum perfectiſſima hiſce ratio-
              <lb/>
            nibus patebit, primo ſicut circulus omnibus planis figuris præcellit, ita quoq; </s>
            <s xml:id="echoid-s2878" xml:space="preserve">ſphæra ſolidas omnes figuras
              <lb/>
            antecellit; </s>
            <s xml:id="echoid-s2879" xml:space="preserve">nam ſicut circulus vnica linea, ſic ſphæra vnica ſuperficie concluditur; </s>
            <s xml:id="echoid-s2880" xml:space="preserve">ſicut in circulo apparet
              <lb/>
            maxima partiũ conſormitas, ac ſimilitudo, qua a medio vniformiter diſtant; </s>
            <s xml:id="echoid-s2881" xml:space="preserve">ita etiam omnes ſphæræ par-
              <lb/>
            tes ab ipſius medio conſimiliter recedunt, vnde etiã ipſius maxima pulchritudo exoritur: </s>
            <s xml:id="echoid-s2882" xml:space="preserve">præterea in neu-
              <lb/>
            tra harum figurarum principium, aut finem eſt aſſignare: </s>
            <s xml:id="echoid-s2883" xml:space="preserve">Inſuper, vtraque eundem ſemper in ſua reuolu-
              <lb/>
            tione locum occupat. </s>
            <s xml:id="echoid-s2884" xml:space="preserve">tandem vtraque eſt omnium figurarum ſibi Iſoperimetrarum maximè capax. </s>
            <s xml:id="echoid-s2885" xml:space="preserve">ſed ne
              <lb/>
            longior ſim, vide Proemium Mecha. </s>
            <s xml:id="echoid-s2886" xml:space="preserve">quæſt. </s>
            <s xml:id="echoid-s2887" xml:space="preserve">Ari otelis, cum noſtra explicatione in libro locorum Mathe-
              <lb/>
            maticorum Ariſtotelis, vbi de admirandis circu@s proprietatibus fuſius diſſeritur. </s>
            <s xml:id="echoid-s2888" xml:space="preserve">porrò ſphæram eſſe cir-
              <lb/>
            culo præſtantiorem hinc patet; </s>
            <s xml:id="echoid-s2889" xml:space="preserve">ille enim ſuperficies eſt duabus tantum dimenſionibus longitudine, & </s>
            <s xml:id="echoid-s2890" xml:space="preserve">la-
              <lb/>
            titudine prædita; </s>
            <s xml:id="echoid-s2891" xml:space="preserve">hæc verò eſt corpus tribus dimenſionibus conſtans, latitudine, longitudine, profundita-
              <lb/>
            te, qua propter omnium figurarum, tum planarum, tum ſolidarum ſphæra obtinet principatum.</s>
            <s xml:id="echoid-s2892" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2893" xml:space="preserve">Vt autem ratio illa deſumpta à capacitate Iſoperimetrarum figurarum probè percipiatur, nonnulla de
              <lb/>
            Iſoperimetris figuris in medium ſunt proferenda.</s>
            <s xml:id="echoid-s2894" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2895" xml:space="preserve">Iſoperimetræ igitur figuræ ſunt, quæ habent æquales ambitus, ſeu circum ferentias, ſiue ſint figuræ pla-
              <lb/>
            næ, ſiue ſolidæ, ideſt, ſuperficies, aut corpora; </s>
            <s xml:id="echoid-s2896" xml:space="preserve">quod, & </s>
            <s xml:id="echoid-s2897" xml:space="preserve">eorum nomen pulchrè indicat ισος, enim græcè,
              <lb/>
            æqualem, ſignificat: </s>
            <s xml:id="echoid-s2898" xml:space="preserve">@εριμετρος autem ambitum valet. </s>
            <s xml:id="echoid-s2899" xml:space="preserve">vbi notandum eſt per figuram, cum Geometris, in-
              <lb/>
            telligendam eſſe aeream, ſeu ſpatium tam planum, quam ſolidum terminatum aliqua peripheria, aut am-
              <lb/>
            bitu, non autem ipſum ambitum ſolum, vt Geometriæ expertes perperam ſolent exiſtimare. </s>
            <s xml:id="echoid-s2900" xml:space="preserve">Cum igitur
              <lb/>
            dicimus duas planas figuras, v. </s>
            <s xml:id="echoid-s2901" xml:space="preserve">g. </s>
            <s xml:id="echoid-s2902" xml:space="preserve">triangulum vnum, & </s>
            <s xml:id="echoid-s2903" xml:space="preserve">quadratum vnum eſſe inuicem Iſoperimetra, in-
              <lb/>
            telligimus duas ſuperficies, vnam triangularem, alteram quadratam habere æqualem ambitum, qui ambi-
              <lb/>
            tus erit linea, eas terminans. </s>
            <s xml:id="echoid-s2904" xml:space="preserve">cum verò dicimus duo corpora eſſe Iſoperimetra, v. </s>
            <s xml:id="echoid-s2905" xml:space="preserve">g. </s>
            <s xml:id="echoid-s2906" xml:space="preserve">cubum vnũ vni ſphæ-
              <lb/>
            ræ eſſe Iſoperimetrum, intelligimus ſpatia eorum ſolida, ſeu eorum ſoliditates habere æquales ambitus,
              <lb/>
            ideſt, terminari æqualibus ſuperficiebus, corpora enim ſuperficiebus terminantur.</s>
            <s xml:id="echoid-s2907" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2908" xml:space="preserve">Aduertendum præterea eſt, duas figuras planam alteram, alteram vero ſolidam, nulla ratione poſſe eſſe
              <lb/>
            mutuo Iſoperimetras, quia cum earum ambitus ſint diuerſi generis, planorum enim ſunt lineæ ambientes,
              <lb/>
            ſolidarum verò ſuperficies, nequeunt inter ipſas reperiri vllæ proportiones, vt conſtat ex definitione ter-
              <lb/>
            tia lib. </s>
            <s xml:id="echoid-s2909" xml:space="preserve">5. </s>
            <s xml:id="echoid-s2910" xml:space="preserve">Elem. </s>
            <s xml:id="echoid-s2911" xml:space="preserve">Euclidis; </s>
            <s xml:id="echoid-s2912" xml:space="preserve">quare neq; </s>
            <s xml:id="echoid-s2913" xml:space="preserve">proportionem æqualitatis inter eas reperire erit, ideſt, linea, & </s>
            <s xml:id="echoid-s2914" xml:space="preserve">ſuper-
              <lb/>
              <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a" number="22">
                <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0045-01"/>
              </figure>
            ficies neq; </s>
            <s xml:id="echoid-s2915" xml:space="preserve">æquales, neq; </s>
            <s xml:id="echoid-s2916" xml:space="preserve">inæqua-
              <lb/>
            les inuicem eſie poſſunt. </s>
            <s xml:id="echoid-s2917" xml:space="preserve">his præ-
              <lb/>
            notatis probandum eſt circulum
              <lb/>
            inter omnes planas figuras Iſo-
              <lb/>
            perimetras ſphæram verò inter
              <lb/>
            ſolidas pariter Iſoperimetras eſ-
              <lb/>
            ſe capaciſſimam. </s>
            <s xml:id="echoid-s2918" xml:space="preserve">Exponãtur pri-
              <lb/>
            mo aliquot planæ figuræ Iſope-
              <lb/>
            rimetræ, quarum prima ſit trian-
              <lb/>
            gulum Iſoſceles, vt in figura vi-
              <lb/>
            des, cuius ſingula latera cõſtent lineolis 5. </s>
            <s xml:id="echoid-s2919" xml:space="preserve">æqualibus, baſis verò 6. </s>
            <s xml:id="echoid-s2920" xml:space="preserve">ſic enim eius perimeter, ſeu ambitus con-
              <lb/>
            tinebit huiuſmodi lineolas 16. </s>
            <s xml:id="echoid-s2921" xml:space="preserve">quarum modulus ſit linea F. </s>
            <s xml:id="echoid-s2922" xml:space="preserve">diuiſa in 16. </s>
            <s xml:id="echoid-s2923" xml:space="preserve">particulas æquales. </s>
            <s xml:id="echoid-s2924" xml:space="preserve">ſecunda figu-
              <lb/>
            ra ſit quadratum, cuius ſingula latera contineant quatuor lineolas æquales prædictis, ſic enim erit eius
              <lb/>
            perimeter 16. </s>
            <s xml:id="echoid-s2925" xml:space="preserve">Tertia ſic circulus, cuius perimeter, vel peripheria compræhendat etiam 16. </s>
            <s xml:id="echoid-s2926" xml:space="preserve">ex prædictis
              <lb/>
            lineolis. </s>
            <s xml:id="echoid-s2927" xml:space="preserve">Cum igitur omnium perimeter ſit 16. </s>
            <s xml:id="echoid-s2928" xml:space="preserve">ſecundum æquales menſuras, erunt omnes tres inuicem I
              <unsure/>
            ſo-
              <lb/>
            perimetræ. </s>
            <s xml:id="echoid-s2929" xml:space="preserve">conſtruximus autem circulum alijs duabus Iſoperimetrũ hac ratione: </s>
            <s xml:id="echoid-s2930" xml:space="preserve">conſtat enim ex demon-
              <lb/>
            ſtratis ab Archimede, quod etiam experimento patere poteſt, circumferẽtiam circuli ad ſuam diametrum
              <lb/>
            habere ferè eandem rationem quam habent 22. </s>
            <s xml:id="echoid-s2931" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s2932" xml:space="preserve">quare per auream Arithmeticæ regulam, reperio ita
              <lb/>
            ſe habere 22. </s>
            <s xml:id="echoid-s2933" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s2934" xml:space="preserve">quemadmodum 16. </s>
            <s xml:id="echoid-s2935" xml:space="preserve">ambitus ſcilicet quaſi circuli, ad 5. </s>
            <s xml:id="echoid-s2936" xml:space="preserve">& </s>
            <s xml:id="echoid-s2937" xml:space="preserve">vnam vndecimam, quare 5. </s>
            <s xml:id="echoid-s2938" xml:space="preserve">& </s>
            <s xml:id="echoid-s2939" xml:space="preserve">
              <lb/>
            vna vndecima ex illis lineolis, erit quæſita diameter. </s>
            <s xml:id="echoid-s2940" xml:space="preserve">huius diametri dimidium eſt 2. </s>
            <s xml:id="echoid-s2941" xml:space="preserve">& </s>
            <s xml:id="echoid-s2942" xml:space="preserve">ſex vndecimę, acce-
              <lb/>
            ptis igitur pro ſemidiametro 2. </s>
            <s xml:id="echoid-s2943" xml:space="preserve">& </s>
            <s xml:id="echoid-s2944" xml:space="preserve">ſex vndecimis ex prædictis lineolis. </s>
            <s xml:id="echoid-s2945" xml:space="preserve">earum interuallo deſcriptus eſt cir-
              <lb/>
            culus alijs duabus figuris Iſoperimeter, iam ſingularum areæ menſurandæ ſunt, vt appareat circulum eſſe
              <lb/>
            earum capaciſſimum, atq; </s>
            <s xml:id="echoid-s2946" xml:space="preserve">adeo maximum.</s>
            <s xml:id="echoid-s2947" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2948" xml:space="preserve">Quemadmodum autem Geometræ aptè lineas æqualibus lineis metiuntur, ita etiam ſuperficies, ſeu pla-
              <lb/>
            nas figuras æqualibus planis, videlicet æqualibus quadratis menſurant, quia, vel teſte Ariſtotele, menſura
              <lb/>
            debet eſſe eiuſdem generis cũ re menſurata, menſuratio trianguli ſic perficitur; </s>
            <s xml:id="echoid-s2949" xml:space="preserve">ducta perpendiculari A D.
              <lb/>
            </s>
            <s xml:id="echoid-s2950" xml:space="preserve">quæ baſim bifariam ſecat, dimidium baſis, quod eſt 3. </s>
            <s xml:id="echoid-s2951" xml:space="preserve">ducitur in perpendiculum A D. </s>
            <s xml:id="echoid-s2952" xml:space="preserve">quod eſt 4. </s>
            <s xml:id="echoid-s2953" xml:space="preserve">vnde pro-
              <lb/>
            ducuntur 12. </s>
            <s xml:id="echoid-s2954" xml:space="preserve">ideſt, 12. </s>
            <s xml:id="echoid-s2955" xml:space="preserve">quadrata æqualia, quorum latera ſunt lineolæ æquales prædictæ, hac autem 12. </s>
            <s xml:id="echoid-s2956" xml:space="preserve">qua-
              <lb/>
            drata conſtituunt aream trianguli, & </s>
            <s xml:id="echoid-s2957" xml:space="preserve">proinde ipſius magnitudinem produnt. </s>
            <s xml:id="echoid-s2958" xml:space="preserve">quod manifeſtius ſit, ſi com-
              <lb/>
            pleatur rectangulum A D B E. </s>
            <s xml:id="echoid-s2959" xml:space="preserve">id enim erit æquale toti triangulo A B C. </s>
            <s xml:id="echoid-s2960" xml:space="preserve">vt figuram contemplanti patere
              <lb/>
            poteſt; </s>
            <s xml:id="echoid-s2961" xml:space="preserve">& </s>
            <s xml:id="echoid-s2962" xml:space="preserve">ex 42. </s>
            <s xml:id="echoid-s2963" xml:space="preserve">primi Elem. </s>
            <s xml:id="echoid-s2964" xml:space="preserve">Euclidis. </s>
            <s xml:id="echoid-s2965" xml:space="preserve">Continet autem hoc rectangulum 12. </s>
            <s xml:id="echoid-s2966" xml:space="preserve">parua quadrata, quæ eſt area
              <lb/>
            trianguli, vt dictum eſt. </s>
            <s xml:id="echoid-s2967" xml:space="preserve">Quadratum autem continet 16. </s>
            <s xml:id="echoid-s2968" xml:space="preserve">quadrata æqualia prædictis: </s>
            <s xml:id="echoid-s2969" xml:space="preserve">quare ipſius area ma-
              <lb/>
            ior eſt area trianguli; </s>
            <s xml:id="echoid-s2970" xml:space="preserve">quoniam quamuis illi ſit Iſoperimetrum magis tamen ad rotũditatem accedit, ideſt,
              <lb/>
            anguli ipſius magis dilatantur, ac proinde euadit capacius, ac maius. </s>
            <s xml:id="echoid-s2971" xml:space="preserve">Circuli menſuratio ſic abſoluitur à
              <lb/>
            Geometris; </s>
            <s xml:id="echoid-s2972" xml:space="preserve">ducunt ſemidiametrum in ſemicircumferentiam, & </s>
            <s xml:id="echoid-s2973" xml:space="preserve">quod producitur eſt circuli area, ſeu </s>
          </p>
        </div>
      </text>
    </echo>
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