Clavius, Christoph, Geometria practica

Table of contents

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[361.] PROBL. 9. PROPOS. 17.
Page: 379 (351)
[362.] PROBL. 10. PROPOS. 18.
Page: 379 (351)
[363.] PROBL. 11. PROPOS. 19.
Page: 380 (352)
[364.] PROBL. 12. PROPOS. 20.
Page: 381 (353)
[365.] THEOR. 9. ROPOS. 21.
Page: 381 (353)
[366.] PROBL. 13. PROPOS. 22.
Page: 382 (354)
[367.] PROBL. 14. PROPOS. 23.
Page: 383 (355)
[368.] PROBL. 15. PROPOS. 24.
Page: 383 (355)
[369.] PROBL. 16. PROPOS. 25.
Page: 384 (356)
[370.] PROBL. 17. PROPOS. 26.
Page: 385 (357)
[371.] COROLLARIVM.
Page: 385 (357)
[372.] PROBL. 18. PROPOS. 27.
Page: 385 (357)
[373.] THEOR. 10. PROPOS. 28.
Page: 386 (358)
[374.] SCHOLIVM.
Page: 388 (360)
[375.] THEOR. 11. PROPOS. 29.
Page: 388 (360)
[376.] SCHOLIVM.
Page: 390 (362)
[377.] THEOR. 12. PROPOS. 30.
Page: 390 (362)
[378.] THEOR. 13. PROPOS. 31.
Page: 392 (364)
[379.] THEOR. 14. PROPOS. 32.
Page: 393 (365)
[380.] PROBL. 19. PROPOS. 33.
Page: 393 (365)
[381.] SCHOLIVM.
Page: 394 (366)
[382.] PROBL. 20. PROPOS. 34.
Page: 394 (366)
[383.] PROBL. 21. PROPOS. 35.
Page: 395 (367)
[384.] PROBL. 22. PROPOS. 36.
Page: 396 (368)
[385.] COROLLARIVM I.
Page: 396 (368)
[386.] COROLLARIVM II.
Page: 397 (369)
[387.] PROBL. 23. PROPOS. 37.
Page: 397 (369)
[388.] PROBL. 24. PROPOS. 38.
Page: 397 (369)
[389.] COROLLARIVM.
Page: 397 (369)
[390.] PROBL. 25. PROPOS. 39.
Page: 398 (370)
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            <s xml:id="echoid-s15607" xml:space="preserve">
              <pb o="334" file="364" n="364" rhead="GEOMETR. PRACT."/>
            cus LF, FC, CG, GB, bifariam, & </s>
            <s xml:id="echoid-s15608" xml:space="preserve">hos rurſus bifariã, & </s>
            <s xml:id="echoid-s15609" xml:space="preserve">c. </s>
            <s xml:id="echoid-s15610" xml:space="preserve">connectemuſq; </s>
            <s xml:id="echoid-s15611" xml:space="preserve">rectas,
              <lb/>
            donec fiat exceſſus minor ex ceſſu H, per ſuperius principium Cardani. </s>
            <s xml:id="echoid-s15612" xml:space="preserve">Quo-
              <lb/>
            niam igitur arcus L B, prima quantitas ſuperat ſecundam, videlicet rectas L F,
              <lb/>
            FC, CG, GB, ſimul exceſſu I; </s>
            <s xml:id="echoid-s15613" xml:space="preserve">Et tertia quantitas, nimirum ſumma rectarum AL,
              <lb/>
            AB, ſuperat quartam, id eſt, ſummam rectarum LD, DE, EB, exceſſu H: </s>
            <s xml:id="echoid-s15614" xml:space="preserve">Eſt que
              <lb/>
            exceſſus I, minor exceſſu H; </s>
            <s xml:id="echoid-s15615" xml:space="preserve">Et prima quantitas, hoc eſt, arcus BL, ponitur non
              <lb/>
            minor, quam tertia ex AB, AL, conflata; </s>
            <s xml:id="echoid-s15616" xml:space="preserve">item tertia AB, AI, maior, quam quar-
              <lb/>
            ta LD, DE, EB: </s>
            <s xml:id="echoid-s15617" xml:space="preserve">erit per 1. </s>
            <s xml:id="echoid-s15618" xml:space="preserve">Lemma, minor proportio arcus BL, primæ quantita-
              <lb/>
            tis ad ſecundam LF, FC, CG, GB q@iam tertiæ quantitatis AL, AB, ad quartam
              <lb/>
            L D, D E, E B; </s>
            <s xml:id="echoid-s15619" xml:space="preserve"> Et permutando minor erit proportio arcus L B, ad A L, A
              <note symbol="a" position="left" xlink:label="note-364-01" xlink:href="note-364-01a" xml:space="preserve">ſchol. 27.
                <lb/>
              quinti.</note>
            ſimul, quam rectarum LF, FC, CG, GB, ſimul ad rectas LD, DE, EB, ſimul. </s>
            <s xml:id="echoid-s15620" xml:space="preserve">Sit
              <lb/>
            ergo vt compoſita ex LF, FC, CG, GB, ad compoſitam ex LD, DE, EB, ita ar-
              <lb/>
            cus BK, ad rectas AL, AB, ſimul: </s>
            <s xml:id="echoid-s15621" xml:space="preserve">Eritque propterea minor etiam proportio ar-
              <lb/>
            cus B L, ad AL, AB, ſimul, quam arcus B L, ad arcum BK; </s>
            <s xml:id="echoid-s15622" xml:space="preserve"> ideo que arcus
              <note symbol="b" position="left" xlink:label="note-364-02" xlink:href="note-364-02a" xml:space="preserve">10. quinti.</note>
            maior erit arcu BL. </s>
            <s xml:id="echoid-s15623" xml:space="preserve">Cum ergo eadem ſit proportio rectarum LF, FC, CG, GB,
              <lb/>
            ſimul ad LD, DE, EB, ſimul, quæ arcus BK, ad AL, AB, ſimul: </s>
            <s xml:id="echoid-s15624" xml:space="preserve">ſintque per 3. </s>
            <s xml:id="echoid-s15625" xml:space="preserve">Lem-
              <lb/>
            ma, rectæ LF, FC, CG, GB, ſimul minores, quam LD, DE, EB, ſimul; </s>
            <s xml:id="echoid-s15626" xml:space="preserve">erit quo-
              <lb/>
            que arcus B K, minor, quam AL, AB, ſimul. </s>
            <s xml:id="echoid-s15627" xml:space="preserve">Multò ergo minor erit arcus BL,
              <lb/>
            duabus AL, AB, ſimul. </s>
            <s xml:id="echoid-s15628" xml:space="preserve">Quare rectæ tangentes AL, AB, ſimul maiores ſunt ar-
              <lb/>
            cu BL, quod erat oſtendendum.</s>
            <s xml:id="echoid-s15629" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15630" xml:space="preserve">
              <emph style="sc">Est</emph>
            autem hæc demonſtratio Cardani admirabilis, & </s>
            <s xml:id="echoid-s15631" xml:space="preserve">non abſimilis illi, qua
              <lb/>
            Eucl, in propoſ. </s>
            <s xml:id="echoid-s15632" xml:space="preserve">12. </s>
            <s xml:id="echoid-s15633" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s15634" xml:space="preserve">9. </s>
            <s xml:id="echoid-s15635" xml:space="preserve">vtitur. </s>
            <s xml:id="echoid-s15636" xml:space="preserve">In vtraque enim infertur concluſio demonſtra-
              <lb/>
            tione affirmatiua ex eius oppoſito, vt patet.</s>
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          </p>
          <p>
            <s xml:id="echoid-s15638" xml:space="preserve">
              <emph style="sc">Attvli</emph>
            hanc demonſtrationem Cardani, non quòd verè Geometrica ſit,
              <lb/>
            niſi principium illud ſuum admittatur, ſed quod ingenioſa ſit & </s>
            <s xml:id="echoid-s15639" xml:space="preserve">acuta. </s>
            <s xml:id="echoid-s15640" xml:space="preserve">Sine ta-
              <lb/>
            men hac demonſtratione concedendum erit, ambitum figuræ circumſcriptæ eſ-
              <lb/>
            ſe inaiorem peripheria circuli propter demonſtrationem Archimedis, cumnihil
              <lb/>
            vnquam in contrarium à quo quam ſit allatum, vt ſupra diximus.</s>
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          </p>
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        <div xml:id="echoid-div958" type="section" level="1" n="339">
          <head xml:id="echoid-head366" xml:space="preserve">THEOR. 2. PROPOS. 2.</head>
          <p>
            <s xml:id="echoid-s15642" xml:space="preserve">CIRCVLORVM diametri inter ſe ſunt, vt circumferentiæ.</s>
            <s xml:id="echoid-s15643" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15644" xml:space="preserve">
              <emph style="sc">Hoc</emph>
            demonſtrauimus nos in libr. </s>
            <s xml:id="echoid-s15645" xml:space="preserve">4. </s>
            <s xml:id="echoid-s15646" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s15647" xml:space="preserve">7. </s>
            <s xml:id="echoid-s15648" xml:space="preserve">num. </s>
            <s xml:id="echoid-s15649" xml:space="preserve">3. </s>
            <s xml:id="echoid-s15650" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s15651" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15652" xml:space="preserve">idem autem
              <lb/>
            hic aliter demonſtrabimus ex Pappo, hoc modo. </s>
            <s xml:id="echoid-s15653" xml:space="preserve">Sint duo circuli A B C D,
              <lb/>
            EFGH, quorum diametri AC, EG. </s>
            <s xml:id="echoid-s15654" xml:space="preserve">Dico eſſe cir-
              <lb/>
              <figure xlink:label="fig-364-01" xlink:href="fig-364-01a" number="255">
                <image file="364-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/364-01"/>
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            cumferentiam ad circumferentiam, vt eſt diameter
              <lb/>
            ad diametrum. </s>
            <s xml:id="echoid-s15655" xml:space="preserve"> Quoniam enim eſt circulus ad
              <note symbol="c" position="left" xlink:label="note-364-03" xlink:href="note-364-03a" xml:space="preserve">2. duodec.</note>
            culum, vt quadratum diametri ad quadratum dia-
              <lb/>
            metri. </s>
            <s xml:id="echoid-s15656" xml:space="preserve"> Vt autem circulus A B C D, ad
              <note symbol="d" position="left" xlink:label="note-364-04" xlink:href="note-364-04a" xml:space="preserve">15. quinti.</note>
            E F G H, ita eſt quadruplum circuli ad quadru-
              <lb/>
            plam circuli. </s>
            <s xml:id="echoid-s15657" xml:space="preserve">Igitur erit quoque quadruplum
              <lb/>
            circuli A B C D, ad quadruplum circuli E F-
              <lb/>
            G H, vt quadratum diametri A C, ad quad atum
              <lb/>
            diametri EG. </s>
            <s xml:id="echoid-s15658" xml:space="preserve">Sed rectangulum ſub diametro AC, & </s>
            <s xml:id="echoid-s15659" xml:space="preserve">recta, quæ circumferentiæ
              <lb/>
            ABCD, ſit æqualis, comprehenſum, quadruplũ eſt circuli ABCD; </s>
            <s xml:id="echoid-s15660" xml:space="preserve">& </s>
            <s xml:id="echoid-s15661" xml:space="preserve">rectangu-
              <lb/>
            lum ſub diametro E G, & </s>
            <s xml:id="echoid-s15662" xml:space="preserve">circumferentia EFGH, quadruplum circuli E F G </s>
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