Clavius, Christoph, In Sphaeram Ioannis de Sacro Bosco commentarius

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          <pb o="99" file="135" n="136" rhead="Ioan. de Sacro Boſco."/>
        </div>
        <div xml:id="echoid-div262" type="section" level="1" n="90">
          <head xml:id="echoid-head94" xml:space="preserve">THEOR. 13. PROPOS. 15.</head>
          <p style="it">
            <s xml:id="echoid-s4802" xml:space="preserve">
              <emph style="sc">Area</emph>
            cuiuslibet corporis planis ſuper ficiebus contenti, & </s>
            <s xml:id="echoid-s4803" xml:space="preserve">circa ſphę
              <lb/>
              <note position="right" xlink:label="note-135-01" xlink:href="note-135-01a" xml:space="preserve">Corpus
                <lb/>
              quodlibet,
                <lb/>
              in qua ſphę
                <lb/>
              ra deſcribi
                <lb/>
              poteſt, cui
                <lb/>
              parallelepi-
                <lb/>
              pedo æqua-
                <lb/>
              le ſit.</note>
            ram aliquam circumſcriptibilis, hoc eſt, à cuius puncto aliquo medio omnes
              <lb/>
            perpendiculares ad baſes eius productæ ſunt æquales, æqualis eſt ſolido re-
              <lb/>
            ctangulo contento ſub una perpendicularium, & </s>
            <s xml:id="echoid-s4804" xml:space="preserve">tertia parte ambitus cor-
              <lb/>
            poris.</s>
            <s xml:id="echoid-s4805" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4806" xml:space="preserve">
              <emph style="sc">Esto</emph>
            corpus planis ſupeificiebus contentum A B C D, circa ſphæram
              <lb/>
            E F G H, cuius centrum I, deſcriptum, in quo ducantur ex I, ad puncta con-
              <lb/>
            tactuum lineę rectæ I E, I F, I G, I H, quæ ad baſes ſolidi erunt perpendicula-
              <lb/>
            res. </s>
            <s xml:id="echoid-s4807" xml:space="preserve">Nam ſi v. </s>
            <s xml:id="echoid-s4808" xml:space="preserve">g. </s>
            <s xml:id="echoid-s4809" xml:space="preserve">per rectam I E, ducatur planum faciens in ſphæra, per propoſ.
              <lb/>
            </s>
            <s xml:id="echoid-s4810" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4811" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4812" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4813" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s4814" xml:space="preserve">circulum E F G H, & </s>
            <s xml:id="echoid-s4815" xml:space="preserve">in baſi rectam A B, tanget circulus
              <lb/>
            E F G H, rectam A B, in puncto E, propterea quòd ſphæra baſim non ſecat,
              <lb/>
              <note position="right" xlink:label="note-135-02" xlink:href="note-135-02a" xml:space="preserve">3. undec.</note>
            ſed tangit. </s>
            <s xml:id="echoid-s4816" xml:space="preserve">Igitur I E, ad rectam A B, perpendicularis erit. </s>
            <s xml:id="echoid-s4817" xml:space="preserve">Eadem ratione, ſi
              <lb/>
            per I E, ducatur aliud planum à priori dif-
              <lb/>
              <figure xlink:label="fig-135-01" xlink:href="fig-135-01a" number="37">
                <image file="135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/135-01"/>
              </figure>
              <note position="right" xlink:label="note-135-03" xlink:href="note-135-03a" xml:space="preserve">18. tertij.</note>
            ferens, fiet alius circulus in ſphęra, & </s>
            <s xml:id="echoid-s4818" xml:space="preserve">alia li
              <lb/>
            nea recta in eadem baſi ſecans rectam A B,
              <lb/>
            in E, ad quã etiam I E, perpẽdicularis erit
              <lb/>
            Ac propterea I E, ad baſim ſolidi per illas
              <lb/>
            rectas ductam perpendicularis erit. </s>
            <s xml:id="echoid-s4819" xml:space="preserve">Nõ ali-
              <lb/>
            ter oſtendemus, rectas I F, I G, I H, ad
              <lb/>
              <note position="right" xlink:label="note-135-04" xlink:href="note-135-04a" xml:space="preserve">4. vndec.</note>
            alias baſes eſſe perpendiculares. </s>
            <s xml:id="echoid-s4820" xml:space="preserve">Sit quo-
              <lb/>
            que ſolidum rectangulum L R, cuius baſis
              <lb/>
            K L M N, ſit æqualis tertiæ parti ambitus
              <lb/>
            corporis A B C D; </s>
            <s xml:id="echoid-s4821" xml:space="preserve">altitudo uero, ſiue per
              <lb/>
            pendicularis L P, æqualis uni perpendicu-
              <lb/>
            lariũ ex centro I, ad baſes corporis ABCD,
              <lb/>
            cadentiũ; </s>
            <s xml:id="echoid-s4822" xml:space="preserve">quæ omnes inter ſe ęquales ſunt
              <lb/>
            ex defi. </s>
            <s xml:id="echoid-s4823" xml:space="preserve">ſphæræ. </s>
            <s xml:id="echoid-s4824" xml:space="preserve">Dico, ſolidum L R, corpori
              <lb/>
            A B C D, æquale eſſe. </s>
            <s xml:id="echoid-s4825" xml:space="preserve">Ducantur enim ex
              <lb/>
            centro I, ad oẽs angulos corporis ABCD,
              <lb/>
            rectę lineę, vt totum corpus in pyramides,
              <lb/>
            ex quibus componitur, diuidatur: </s>
            <s xml:id="echoid-s4826" xml:space="preserve">quarum
              <lb/>
            quidem pyramidum baſes e
              <unsure/>
            ędem ſunt, quę
              <lb/>
            corporis, vertex autem communis centrum I. </s>
            <s xml:id="echoid-s4827" xml:space="preserve">Quoniam igitur (per præceden
              <lb/>
            tem propoſ.) </s>
            <s xml:id="echoid-s4828" xml:space="preserve">quælibet harum pyramidum æqualis eſt ſolido rectangulo ſub
              <lb/>
            perpendiculari L P, quæ ſingulis perpendicularibus corporis A B C D, æqua-
              <lb/>
            lis ponitur, & </s>
            <s xml:id="echoid-s4829" xml:space="preserve">tertia parte ſuæ baſis contento; </s>
            <s xml:id="echoid-s4830" xml:space="preserve">Si fiant tot ſolida rectangula,
              <lb/>
            quot ſunt pyramides, erunt omnia hęc ſimul æqualia ſolido rectangulo L R.
              <lb/>
            </s>
            <s xml:id="echoid-s4831" xml:space="preserve">(Si enim rectangulum K L M N, diuidatur in tot rectangula, quot baſes ſunt
              <lb/>
            in ſolido propoſito, ita ut primum æquale ſit tertię parti unius baſis, & </s>
            <s xml:id="echoid-s4832" xml:space="preserve">ſe-
              <lb/>
            cundum tertiæ parti alterius, & </s>
            <s xml:id="echoid-s4833" xml:space="preserve">ita deinceps, quandoquidem totum rectangu-
              <lb/>
            lum K L M N, æquale ponitur tertię parti totius ambitus ſolidi, intelligan-
              <lb/>
            tur autem ſuper illa rectangula conſtitui parallelepipeda; </s>
            <s xml:id="echoid-s4834" xml:space="preserve">erunt omnia ſimul
              <lb/>
            æqualia parallelepipedo L R.) </s>
            <s xml:id="echoid-s4835" xml:space="preserve">Cum ergo ſingula parallelepipeda ſingulis py-
              <lb/>
            ramidibus ſint ęqualia, per propoſ. </s>
            <s xml:id="echoid-s4836" xml:space="preserve">pręcedentem; </s>
            <s xml:id="echoid-s4837" xml:space="preserve">erunt quoque omnes </s>
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