Clavius, Christoph, Geometria practica

Table of contents

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[331.] FINIS LIBRI SEPTIMI.
Page: 359 (329)
[332.] GEOMETRIÆ PRACTICÆ LIBER OCTAVVS.
Page: 360 (330)
[333.] Varia Theoremata, ac problemata Geometrica demonſtrans.
Page: 360 (330)
[334.] THEOR. 1. PROPOS. 1.
Page: 360 (330)
[335.] SCHOLIVM.
Page: 361 (331)
[336.] LEMMA I.
Page: 361 (331)
[337.] LEMMA II.
Page: 362 (332)
[338.] EEMMA III.
Page: 362 (332)
[339.] THEOR. 2. PROPOS. 2.
Page: 364 (334)
[340.] SCHOLIVM.
Page: 365 (335)
[341.] THEOR. 3. PROPOS. 3.
Page: 365 (335)
[342.] COROLLARIVM.
Page: 366 (336)
[343.] PROBL. 1. PROPOS. 4.
Page: 366 (336)
[344.] PROBL. 2. PROPOS. 5.
Page: 367 (339)
[345.] ALITER.
Page: 367 (339)
[346.] PROBL. 3. PROPOS. 6.
Page: 367 (339)
[347.] THEOR. 4. PROPOS. 7.
Page: 368 (340)
[348.] SCHOLIVM.
Page: 368 (340)
[349.] PROBL. 4. PROPOS. 8.
Page: 369 (341)
[350.] PROBL. 5. PROPOS. 9.
Page: 370 (342)
[351.] THEOR. 5. PROPOS. 10.
Page: 371 (343)
[352.] THEOR. 6. PROPOS. 11.
Page: 371 (343)
[353.] COROLLARIVM.
Page: 372 (344)
[354.] THEOR. 7. PROPOS. 12.
Page: 372 (344)
[355.] PROBL. 6. PROPOS. 13.
Page: 372 (344)
[356.] PROBL. 7. PROPOS. 14.
Page: 373 (345)
[357.] THEOR. 8. PROPOS. 15.
Page: 376 (348)
[358.] PROBL. 8. PROPOS. 16.
Page: 377 (349)
[359.] COROLLARIVM.
Page: 378 (350)
[360.] SCHOLIVM.
Page: 379 (351)
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            <s xml:id="echoid-s14480" xml:space="preserve">
              <emph style="sc">Esto</emph>
            corpus planis ſuperficiebus contentum A B C D, circa ſphæram
              <lb/>
            EFGH, cuius centrum I, deſcriptum, in quo ducantur ex I, ad puncta conta-
              <lb/>
            ctuum lineæ rectæ IE, IF, IG, IH, quæ ad baſes ſolidi erunt perpendiculares. </s>
            <s xml:id="echoid-s14481" xml:space="preserve">Nam
              <lb/>
            ſi verbi gratia per rectam IE, ducatur planum faciens in ſphæra, per propoſ. </s>
            <s xml:id="echoid-s14482" xml:space="preserve">1.
              <lb/>
            </s>
            <s xml:id="echoid-s14483" xml:space="preserve">
              <note symbol="a" position="left" xlink:label="note-338-01" xlink:href="note-338-01a" xml:space="preserve">3. vndec.</note>
            lib. </s>
            <s xml:id="echoid-s14484" xml:space="preserve">1. </s>
            <s xml:id="echoid-s14485" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s14486" xml:space="preserve">circulum EFGH, & </s>
            <s xml:id="echoid-s14487" xml:space="preserve">in baſi rectam AB; </s>
            <s xml:id="echoid-s14488" xml:space="preserve">tanget circulus EFGH, rectam A B, in puncto E, propterea quod ſphæra baſim non ſecat, ſed tangit.
              <lb/>
            </s>
            <s xml:id="echoid-s14489" xml:space="preserve">
              <note symbol="b" position="left" xlink:label="note-338-02" xlink:href="note-338-02a" xml:space="preserve">18. tertii.</note>
            Igitur IE, ad rectam AB, perpendicularis erit. </s>
            <s xml:id="echoid-s14490" xml:space="preserve">Eadem ratione, ſi per I E, duca- tur aliud planum, à priori differens, fiet alius circulus in ſphæra, & </s>
            <s xml:id="echoid-s14491" xml:space="preserve">alia linea recta
              <lb/>
            in eadem baſi ſecans rectam A B, in E; </s>
            <s xml:id="echoid-s14492" xml:space="preserve">ad quam et-
              <lb/>
              <figure xlink:label="fig-338-01" xlink:href="fig-338-01a" number="230">
                <image file="338-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/338-01"/>
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              <note symbol="c" position="left" xlink:label="note-338-03" xlink:href="note-338-03a" xml:space="preserve">4. vndec.</note>
            iam I E, perpendicularis erit: </s>
            <s xml:id="echoid-s14493" xml:space="preserve"> Ac propterea IE, ad baſim ſolidi per illas rectas ductam perpendicularis
              <lb/>
            erit. </s>
            <s xml:id="echoid-s14494" xml:space="preserve">Non aliter oſtendemus, rectas IF, IG, IH, ad alias
              <lb/>
            baſes eſſe perpendiculares. </s>
            <s xml:id="echoid-s14495" xml:space="preserve">Sit quo que ſolidum re-
              <lb/>
            ctangulum L R, cuius baſes KLMN, ſit æqualis ter-
              <lb/>
            tiæ parti ambitus corporis ABCD; </s>
            <s xml:id="echoid-s14496" xml:space="preserve">altitudo verò ſi-
              <lb/>
            ue perpendicularis L P, æqualis vni perpendicula-
              <lb/>
            rium ex centro I, ad baſes corporis A B C D, caden-
              <lb/>
            tium; </s>
            <s xml:id="echoid-s14497" xml:space="preserve">quæ omnes inter ſe æquales ſunt ex defin.
              <lb/>
            </s>
            <s xml:id="echoid-s14498" xml:space="preserve">ſphæræ. </s>
            <s xml:id="echoid-s14499" xml:space="preserve">Dico ſolidum LR, corpori ABCD, æquale
              <lb/>
            eſſe. </s>
            <s xml:id="echoid-s14500" xml:space="preserve">Ducantur enim ex centro I, ad omnes angulos
              <lb/>
            corporis ABCD, rectælineæ, vt totum corpus in py-
              <lb/>
            ramides, ex quibus componitur, diuidatur: </s>
            <s xml:id="echoid-s14501" xml:space="preserve">quarum
              <lb/>
            quidem pyramidum baſes eædem ſunt quæ corpo-
              <lb/>
              <note symbol="d" position="left" xlink:label="note-338-04" xlink:href="note-338-04a" xml:space="preserve">14. hui{us}.</note>
            ris, vertex autem communis centrum I. </s>
            <s xml:id="echoid-s14502" xml:space="preserve"> Quoniam igitur quælibet harum pyramidum æqualis eſt ſolido
              <lb/>
            rectangulo ſub perpendiculari L P, quæ ſingulis perpendicularibus corporis
              <lb/>
            ABCD, æqualis ponitur, & </s>
            <s xml:id="echoid-s14503" xml:space="preserve">tertia parte ſuæ baſis contento; </s>
            <s xml:id="echoid-s14504" xml:space="preserve">Si fiant tot ſolida
              <lb/>
            rectangula, quot ſunt pyramides, erunt omnia hæc ſimul æqualia ſolido rectan-
              <lb/>
            gulo LR. </s>
            <s xml:id="echoid-s14505" xml:space="preserve">(Si enim rectangulum K L M N, diuidatur in tot rectangula, quot
              <lb/>
            baſes ſunt in ſolido propoſito, ita vt primum æquale ſit tertiæ parti vnius baſis,
              <lb/>
            & </s>
            <s xml:id="echoid-s14506" xml:space="preserve">ſecundum tertiæ parti alterius, & </s>
            <s xml:id="echoid-s14507" xml:space="preserve">ita deinceps, quando quidem totum rectan-
              <lb/>
            gulum K L M N, æquale ponitur tertiæ parti totius ambitus ſolidi; </s>
            <s xml:id="echoid-s14508" xml:space="preserve">intelligan-
              <lb/>
            tur autem ſuper illa rectangula conſtitui parallelepipeda; </s>
            <s xml:id="echoid-s14509" xml:space="preserve">erunt omnia ſimul
              <lb/>
              <note symbol="e" position="left" xlink:label="note-338-05" xlink:href="note-338-05a" xml:space="preserve">14. hui{us}.</note>
            æqualia parallelepipedo L R.) </s>
            <s xml:id="echoid-s14510" xml:space="preserve"> Cum ergo ſingula parallelepipeda ſingulis pyramidibus ſint æqualia; </s>
            <s xml:id="echoid-s14511" xml:space="preserve">erunt quo que omnes pyramides, nempe corpus
              <lb/>
            A B C D, ex illis compoſitum, æquale ſolido rectangulo L R. </s>
            <s xml:id="echoid-s14512" xml:space="preserve">Quamobrem
              <lb/>
            area cuiuslibet corporis planis ſuperficiebus contenti, &</s>
            <s xml:id="echoid-s14513" xml:space="preserve">c. </s>
            <s xml:id="echoid-s14514" xml:space="preserve">quod demonſtran-
              <lb/>
            dum erat.</s>
            <s xml:id="echoid-s14515" xml:space="preserve"/>
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          <head xml:id="echoid-head336" xml:space="preserve">THEOR. 14. PROPOS. 16.</head>
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            <s xml:id="echoid-s14516" xml:space="preserve">AREA cuiuslibet ſphæræ æqualis eſt ſolido rectangulo comprehenſo
              <lb/>
              <note position="left" xlink:label="note-338-06" xlink:href="note-338-06a" xml:space="preserve">Sphæræ quæ-
                <lb/>
              libet cui pa-
                <lb/>
              rallelepipedo
                <lb/>
              ſit æqualis.</note>
            ſub ſemidiametro ſphæræ, & </s>
            <s xml:id="echoid-s14517" xml:space="preserve">tertia parte ambitus ſphæræ.</s>
            <s xml:id="echoid-s14518" xml:space="preserve"/>
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            <s xml:id="echoid-s14519" xml:space="preserve">
              <emph style="sc">Esto</emph>
            ſphæra ABC, cuius centrum D, ſemidiameter AD: </s>
            <s xml:id="echoid-s14520" xml:space="preserve">Solidum autem
              <lb/>
            rectangulum E, contentũ ſub ſemidiametro AD, & </s>
            <s xml:id="echoid-s14521" xml:space="preserve">tertia parte ambitus </s>
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