Clavius, Christoph, Geometria practica

Table of contents

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[291.] THEOR. 1. PROPOS. 1.
Page: 322 (292)
[292.] PROBL. 2. PROPOS. 2.
Page: 323 (293)
[293.] THEOR. 3. PROPOS. 3.
Page: 324 (294)
[294.] THEOR. 4. PROPOS. 4.
Page: 324 (294)
[295.] THEOR. 5. PROPOS. 5.
Page: 325 (295)
[296.] THEOR. 6. PROPOS. 6.
Page: 326 (296)
[297.] PROBL. 1. PROPOS. 7.
Page: 327 (297)
[298.] SCHOLIVM.
Page: 327 (297)
[299.] THEOR. 7. PROPOS. 8.
Page: 327 (297)
[300.] THEOR. 8. PROPOS. 9.
Page: 328 (298)
[301.] PROBL. 2. PROPOS. 10.
Page: 329 (299)
[302.] THEOR. 9. PROPOS. 11.
Page: 330 (300)
[303.] THEOR. 10. PROPOS. 12.
Page: 333 (303)
[304.] SCHOLIVM.
Page: 334 (304)
[305.] THEOR. 11. PROPOS. 13.
Page: 336 (306)
[306.] COROLLARIVM.
Page: 336 (306)
[307.] THEOR. 12. PROPOS. 14.
Page: 337 (307)
[308.] THEOR. 13. PROPOS. 15.
Page: 337 (307)
[309.] THEOR. 14. PROPOS. 16.
Page: 338 (308)
[310.] THEOR. 15. PROPOS. 17.
Page: 340 (310)
[311.] COROLLARIVM.
Page: 341 (311)
[312.] THEOR. 16. PROPOS. 18.
Page: 341 (311)
[313.] THEOR. 17. PROPOS. 19.
Page: 343 (313)
[314.] SCHOLIVM.
Page: 343 (313)
[315.] PROBL. 3. PROPOS. 20.
Page: 343 (313)
[316.] PROBL. 4. PROPOS. 21.
Page: 344 (314)
[317.] SCHOLIVM.
Page: 346 (316)
[318.] PROBL. 5. PROPOS. 22.
Page: 346 (316)
[319.] SCHOLIVM.
Page: 347 (317)
[320.] APPENDIX.
Page: 347 (317)
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          <p>
            <s xml:id="echoid-s17126" xml:space="preserve">
              <pb o="364" file="392" n="392" rhead="GEOMETR. PRACT."/>
            pendicularis PQ, in Q, iungantur rectæ QN, MQ. </s>
            <s xml:id="echoid-s17127" xml:space="preserve">Dicit igitur, in Iſoſcele MNQ,
              <lb/>
            vtrumlibet angulorum N, Q, triplum eſſe anguli M. </s>
            <s xml:id="echoid-s17128" xml:space="preserve">quod falſum eſſe, hincin-
              <lb/>
            telligi poteſt. </s>
            <s xml:id="echoid-s17129" xml:space="preserve">Demiſſa perpendiculari QR, pro ſinu arcus QN, vel anguli N,
              <lb/>
            poſito ſinu toto MQ, vel MN, 10000000. </s>
            <s xml:id="echoid-s17130" xml:space="preserve"> Quoniam latus DN, potentia
              <note symbol="a" position="left" xlink:label="note-392-01" xlink:href="note-392-01a" xml:space="preserve">12. quarti-
                <lb/>
              dectmi.</note>
            ſquitertium eſt perpendicularis D O, ſi fiat vt 4. </s>
            <s xml:id="echoid-s17131" xml:space="preserve">ad 3. </s>
            <s xml:id="echoid-s17132" xml:space="preserve">ita 100000000000000.
              <lb/>
            </s>
            <s xml:id="echoid-s17133" xml:space="preserve">quadratum lateris DN, ad aliud reperietur quadratum DO, 75000000000000. </s>
            <s xml:id="echoid-s17134" xml:space="preserve">
              <lb/>
              <note symbol="b" position="left" xlink:label="note-392-02" xlink:href="note-392-02a" xml:space="preserve">ſchol. 4.
                <lb/>
              ſecundi.</note>
            quod cum ſit quadruplum quadrati P O, vel Q R, erit quadratum Q R, 18750000000000. </s>
            <s xml:id="echoid-s17135" xml:space="preserve">ipſumque latus QR, erit 4330127. </s>
            <s xml:id="echoid-s17136" xml:space="preserve">verò minus, vel 4330128.
              <lb/>
            </s>
            <s xml:id="echoid-s17137" xml:space="preserve">vero maius, cui in tabula ſinuum (adhibita parte proportionali) reſpondent
              <lb/>
            grad. </s>
            <s xml:id="echoid-s17138" xml:space="preserve">25. </s>
            <s xml:id="echoid-s17139" xml:space="preserve">min. </s>
            <s xml:id="echoid-s17140" xml:space="preserve">39. </s>
            <s xml:id="echoid-s17141" xml:space="preserve">ſec. </s>
            <s xml:id="echoid-s17142" xml:space="preserve">32. </s>
            <s xml:id="echoid-s17143" xml:space="preserve">pro arcu QN, vel angulo MNQ, quo ablato ex duobus
              <lb/>
            rectis, ſiue ex grad. </s>
            <s xml:id="echoid-s17144" xml:space="preserve">180. </s>
            <s xml:id="echoid-s17145" xml:space="preserve">reliqua erit ſumma angulorum æqualium ad baſem QN,
              <lb/>
            grad. </s>
            <s xml:id="echoid-s17146" xml:space="preserve">154. </s>
            <s xml:id="echoid-s17147" xml:space="preserve">min. </s>
            <s xml:id="echoid-s17148" xml:space="preserve">20. </s>
            <s xml:id="echoid-s17149" xml:space="preserve">ſec. </s>
            <s xml:id="echoid-s17150" xml:space="preserve">28. </s>
            <s xml:id="echoid-s17151" xml:space="preserve">atque idcirco vterque complectetur grad. </s>
            <s xml:id="echoid-s17152" xml:space="preserve">77. </s>
            <s xml:id="echoid-s17153" xml:space="preserve">min. </s>
            <s xml:id="echoid-s17154" xml:space="preserve">10. </s>
            <s xml:id="echoid-s17155" xml:space="preserve">
              <lb/>
            ſec. </s>
            <s xml:id="echoid-s17156" xml:space="preserve">14. </s>
            <s xml:id="echoid-s17157" xml:space="preserve">qui maior eſt, quam triplus anguli M N Q. </s>
            <s xml:id="echoid-s17158" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s17159" xml:space="preserve">25. </s>
            <s xml:id="echoid-s17160" xml:space="preserve">min. </s>
            <s xml:id="echoid-s17161" xml:space="preserve">39. </s>
            <s xml:id="echoid-s17162" xml:space="preserve">ſec. </s>
            <s xml:id="echoid-s17163" xml:space="preserve">32. </s>
            <s xml:id="echoid-s17164" xml:space="preserve">cum
              <lb/>
            hic angulus triplicatus efficiat tantummodo grad. </s>
            <s xml:id="echoid-s17165" xml:space="preserve">76. </s>
            <s xml:id="echoid-s17166" xml:space="preserve">min. </s>
            <s xml:id="echoid-s17167" xml:space="preserve">58. </s>
            <s xml:id="echoid-s17168" xml:space="preserve">ſec. </s>
            <s xml:id="echoid-s17169" xml:space="preserve">36. </s>
            <s xml:id="echoid-s17170" xml:space="preserve">Falſum
              <lb/>
            ergo eſt, quod Candalla nititur probare. </s>
            <s xml:id="echoid-s17171" xml:space="preserve">Paralogiſmos tum Caroli Mariani,
              <lb/>
            tum Candallæ, quos committunt, non eſt huius loci manifeſtare: </s>
            <s xml:id="echoid-s17172" xml:space="preserve">ſatis nobis
              <lb/>
            eſt, indicaſſe eos non rectè deſcripſiſſe heptagonũ æquilaterũ, & </s>
            <s xml:id="echoid-s17173" xml:space="preserve">æquiangulũ.</s>
            <s xml:id="echoid-s17174" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1057" type="section" level="1" n="378">
          <head xml:id="echoid-head405" xml:space="preserve">THEOR. 13. PROPOS. 31.</head>
          <p>
            <s xml:id="echoid-s17175" xml:space="preserve">OCTOGONVM æquilaterum & </s>
            <s xml:id="echoid-s17176" xml:space="preserve">æquiangulum circulo inſcriptum
              <lb/>
            medio loco proportionale eſt inter quadratum eidem circulo circũ-
              <lb/>
            ſcriptum, & </s>
            <s xml:id="echoid-s17177" xml:space="preserve">quadratum inſcriptum.</s>
            <s xml:id="echoid-s17178" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s17179" xml:space="preserve">
              <emph style="sc">Hoc</emph>
            Theorema eſt Orontij, quod facilè ita demonſtrabitur. </s>
            <s xml:id="echoid-s17180" xml:space="preserve">Sit circu-
              <lb/>
            lus ABCD, cuius centrum E; </s>
            <s xml:id="echoid-s17181" xml:space="preserve">duæ diametri AC, BD, ſecantes ſe ſe in E, ad angu-
              <lb/>
            los rectos. </s>
            <s xml:id="echoid-s17182" xml:space="preserve">Iunctis ergo rectis AB, B C, C D, D A, erit quadratum circulo inſcri-
              <lb/>
            ptum ABCD, vt ex demonſtratione propoſ. </s>
            <s xml:id="echoid-s17183" xml:space="preserve">6. </s>
            <s xml:id="echoid-s17184" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s17185" xml:space="preserve">4. </s>
            <s xml:id="echoid-s17186" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s17187" xml:space="preserve">conſtat. </s>
            <s xml:id="echoid-s17188" xml:space="preserve">Ducantur
              <lb/>
              <figure xlink:label="fig-392-01" xlink:href="fig-392-01a" number="284">
                <image file="392-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/392-01"/>
              </figure>
            quoque per A, B, C, D, perpendiculares ad diame-
              <lb/>
            tros coeuntes in F, G, H, I, eritque quadratum cir-
              <lb/>
            cumſcriptum F G H I, vt patet ex demonſtratione
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s17189" xml:space="preserve">7. </s>
            <s xml:id="echoid-s17190" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s17191" xml:space="preserve">4. </s>
            <s xml:id="echoid-s17192" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s17193" xml:space="preserve">Ductis autem diametris
              <lb/>
            FH, GI, ſecabuntur quadrantes AB, BC, CD,
              <note symbol="c" position="left" xlink:label="note-392-03" xlink:href="note-392-03a" xml:space="preserve">27. tertij.</note>
            bifariam; </s>
            <s xml:id="echoid-s17194" xml:space="preserve">propterea quod anguli in centro ſunt o-
              <lb/>
              <note symbol="d" position="left" xlink:label="note-392-04" xlink:href="note-392-04a" xml:space="preserve">ſchol. 34.
                <lb/>
              primi.</note>
            mnes æquales, nimirum ſemirecti: </s>
            <s xml:id="echoid-s17195" xml:space="preserve"> ac proinde &</s>
            <s xml:id="echoid-s17196" xml:space="preserve"> latera quadrati inſcripti diuiſa erunt bifariam, & </s>
            <s xml:id="echoid-s17197" xml:space="preserve">ad
              <lb/>
              <note symbol="e" position="left" xlink:label="note-392-05" xlink:href="note-392-05a" xml:space="preserve">ſchol. 27.
                <lb/>
              tertij.</note>
            angulos rectos. </s>
            <s xml:id="echoid-s17198" xml:space="preserve">Et ſi iungantur rectæ, AK, KB, &</s>
            <s xml:id="echoid-s17199" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s17200" xml:space="preserve">deſcriptum erit octogonum intra circulum. </s>
            <s xml:id="echoid-s17201" xml:space="preserve">Dico
              <lb/>
            ita eſſe quadratum exterius ad octogonum, vt o-
              <lb/>
            ctogonum ad quadratum interius. </s>
            <s xml:id="echoid-s17202" xml:space="preserve">Quoniam enim
              <lb/>
            triangula AEF, EAL, ęquiangula ſunt, quod rectos
              <lb/>
            habeant angulos, & </s>
            <s xml:id="echoid-s17203" xml:space="preserve">ſemirectos: </s>
            <s xml:id="echoid-s17204" xml:space="preserve"> Erit E F, ad F A, hoc eſt, ad EK, (eſt namque EK, ipſi EA, hoc eſt, ipſi AF, æqualis) vt EA, hoc eſt,
              <lb/>
              <note symbol="f" position="left" xlink:label="note-392-06" xlink:href="note-392-06a" xml:space="preserve">4. ſexti.</note>
            vt EK, ad AL, hoc eſt, ad EL, quod AL, EL, ſint æquales, propter angulos
              <note symbol="g" position="left" xlink:label="note-392-07" xlink:href="note-392-07a" xml:space="preserve">6. primi.</note>
            mirectos A, E, intriangulo AEL. </s>
            <s xml:id="echoid-s17205" xml:space="preserve">Sunt ergo tres rectę EF, EK, EL, continue pro-
              <lb/>
            portio les. </s>
            <s xml:id="echoid-s17206" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s17207" xml:space="preserve">triangula AEF, AEK, AEL, continue erunt proportiona-
              <lb/>
              <note symbol="h" position="left" xlink:label="note-392-08" xlink:href="note-392-08a" xml:space="preserve">1. ſexti.</note>
            lia: </s>
            <s xml:id="echoid-s17208" xml:space="preserve"> na baſibus EF; </s>
            <s xml:id="echoid-s17209" xml:space="preserve">EK, EL, ſint proportionalia; </s>
            <s xml:id="echoid-s17210" xml:space="preserve"> Ac proinde & </s>
            <s xml:id="echoid-s17211" xml:space="preserve">eorũ
              <note symbol="i" position="left" xlink:label="note-392-09" xlink:href="note-392-09a" xml:space="preserve">15 quinti@</note>
            </s>
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