Clavius, Christoph, Geometria practica

Table of contents

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[281.] PROBL. 17. PROPOS. 22.
Page: 319 (289)
[282.] FINIS LIBRI SEXTI.
Page: 320 (290)
[283.] GEOMETRIÆ PRACTICÆ LIBER SEPTIMVS.
Page: 321 (291)
[284.] De figuris Iſoperimetris diſputans: cui Appendicis loco annectitur breuis de circulo per lineas quadrando tractatiuncula.
Page: 321 (291)
[285.] DEFINITIONES.
Page: 321 (291)
[286.] I.
Page: 321 (291)
[287.] II.
Page: 322 (292)
[288.] III.
Page: 322 (292)
[289.] IIII.
Page: 322 (292)
[290.] V.
Page: 322 (292)
[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)
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              <pb o="359" file="387" n="387" rhead="LIBER OCTAVVS."/>
            ſecans diametrum in G, ad rectos angulos, iunganturque ad extrema diametri
              <lb/>
              <figure xlink:label="fig-387-01" xlink:href="fig-387-01a" number="279">
                <image file="387-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/387-01"/>
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            rectæ BC, BA. </s>
            <s xml:id="echoid-s16768" xml:space="preserve">Dico tamangulos CBF, CBG,
              <lb/>
            quam (producta F B, ad H,) angulos ABH,
              <lb/>
            ABG, eſſe æquales. </s>
            <s xml:id="echoid-s16769" xml:space="preserve"> Quoniam enim
              <note symbol="a" position="right" xlink:label="note-387-01" xlink:href="note-387-01a" xml:space="preserve">32. tertij.</note>
            C B F, angulo B A C, in alterno ſegmento æ-
              <lb/>
            qualis eſt. </s>
            <s xml:id="echoid-s16770" xml:space="preserve"> & </s>
            <s xml:id="echoid-s16771" xml:space="preserve">angulus C B D, eidem
              <note symbol="b" position="right" xlink:label="note-387-02" xlink:href="note-387-02a" xml:space="preserve">27. tertij.</note>
            B A C, æqualis, ob arcus æquales CB, CD; </s>
            <s xml:id="echoid-s16772" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-387-03" xlink:href="note-387-03a" xml:space="preserve">8. ſexti.</note>
            (vel etiam angulus CBG, angulo BAC, æqua-
              <lb/>
            lis eſt; </s>
            <s xml:id="echoid-s16773" xml:space="preserve">quod BG, in triangulo rectangulo AB-
              <lb/>
            C, ad baſem AC, perpendicularis ſit) erunt an-
              <lb/>
            guli CBF, CBG, inter ſe quoque æquales.
              <lb/>
            </s>
            <s xml:id="echoid-s16774" xml:space="preserve">Producta autem CB, ad I, ſi ex rectis angulis
              <lb/>
            ABC, ABI, tollantur æquales CBG, HBI,
              <note symbol="d" position="right" xlink:label="note-387-04" xlink:href="note-387-04a" xml:space="preserve">15. primi.</note>
            (cum enim CBF, æqualis ſit angulo H B I, ad
              <lb/>
            verticem: </s>
            <s xml:id="echoid-s16775" xml:space="preserve">& </s>
            <s xml:id="echoid-s16776" xml:space="preserve">angulus CBG, angulo CBF, o-
              <lb/>
            ſtenſus æqualis; </s>
            <s xml:id="echoid-s16777" xml:space="preserve">erit quo que angulus C B G,
              <lb/>
            angulo H B I, æqualis.) </s>
            <s xml:id="echoid-s16778" xml:space="preserve">erunt quo que reliqui
              <lb/>
            anguli ABG, ABH, inter ſe æquales. </s>
            <s xml:id="echoid-s16779" xml:space="preserve">quod eſt
              <lb/>
            primum.</s>
            <s xml:id="echoid-s16780" xml:space="preserve"/>
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            <s xml:id="echoid-s16781" xml:space="preserve">
              <emph style="sc">Deinde</emph>
            ducatur recta FN, ſecans circu-
              <lb/>
            lum in K, L, ductiſque rectis KGO, LGM, per
              <lb/>
            G, iungantur tam rectæ K C, K A, quam L C,
              <lb/>
            L A, ad extrema diametri. </s>
            <s xml:id="echoid-s16782" xml:space="preserve">Dico rurſus, tam angulos CLF, CLG, quam ALG,
              <lb/>
            ALN: </s>
            <s xml:id="echoid-s16783" xml:space="preserve">Item tam CKF, CKG, quam AKG, AKL, eſſe æquales. </s>
            <s xml:id="echoid-s16784" xml:space="preserve">Ductis enim ex
              <lb/>
            centro rectis EB, EK; </s>
            <s xml:id="echoid-s16785" xml:space="preserve"> erit angulus EBF, rectus: </s>
            <s xml:id="echoid-s16786" xml:space="preserve"> Igitur erit FB, media
              <note symbol="e" position="right" xlink:label="note-387-05" xlink:href="note-387-05a" xml:space="preserve">18. tertij.</note>
            tionalis inter EF, FG: </s>
            <s xml:id="echoid-s16787" xml:space="preserve"> Ideoque rectangulum ſub EF, FG, quadrato ex FB,
              <note symbol="f" position="right" xlink:label="note-387-06" xlink:href="note-387-06a" xml:space="preserve">coroll. 8.
                <lb/>
              ſexti.</note>
            quale erit; </s>
            <s xml:id="echoid-s16788" xml:space="preserve"> Eſt autem eidem quadrato æquale quo que rectangulum ſub LF, FK. </s>
            <s xml:id="echoid-s16789" xml:space="preserve">Igitur rectangulum ſub EF, FG, rectangulo ſub L F, F K, æquale erit: </s>
            <s xml:id="echoid-s16790" xml:space="preserve">
              <note symbol="g" position="right" xlink:label="note-387-07" xlink:href="note-387-07a" xml:space="preserve">17. ſexti.</note>
            pro inde erit vt EF, prima ad FK, ſecundam, ita LF, tertia ad FG, quartam. </s>
            <s xml:id="echoid-s16791" xml:space="preserve">Qua-
              <lb/>
              <note symbol="h" position="right" xlink:label="note-387-08" xlink:href="note-387-08a" xml:space="preserve">36. tertij.</note>
            re cum triangula EFK, LFG, habeant latera circa communem angulum F, pro-
              <lb/>
              <note symbol="i" position="right" xlink:label="note-387-09" xlink:href="note-387-09a" xml:space="preserve">16. ſexti.</note>
            portionalia; </s>
            <s xml:id="echoid-s16792" xml:space="preserve"> erunt anguli FEK, FLG, homologis lateribus FK, FG,
              <note symbol="k" position="right" xlink:label="note-387-10" xlink:href="note-387-10a" xml:space="preserve">6. ſexti.</note>
            æquales. </s>
            <s xml:id="echoid-s16793" xml:space="preserve"> Eſt autem angulus FEK, in centro anguli CIK, ad
              <note symbol="l" position="right" xlink:label="note-387-11" xlink:href="note-387-11a" xml:space="preserve">20. tertij.</note>
            (cum habeãt eandem baſem C K,) duplus. </s>
            <s xml:id="echoid-s16794" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s16795" xml:space="preserve">angulus FLG, eiuſdem an-
              <lb/>
            guli C L K, duplus erit; </s>
            <s xml:id="echoid-s16796" xml:space="preserve">ac proinde angulus F L G, ſectus erit bifariam à recta
              <lb/>
            LC, hoc eſt, anguli CLF, CLG, æquales erunt. </s>
            <s xml:id="echoid-s16797" xml:space="preserve">Producta autem CL, ad P, ſi ex
              <lb/>
            rectis angulis ALC, ALP, demantur æquales anguli CLG, P L N, (Cum enim
              <lb/>
            CLF, oſtenſus ſit æqualis angulo CLG, & </s>
            <s xml:id="echoid-s16798" xml:space="preserve">CLF, & </s>
            <s xml:id="echoid-s16799" xml:space="preserve">æqualis ſitangulo
              <note symbol="m" position="right" xlink:label="note-387-12" xlink:href="note-387-12a" xml:space="preserve">15. primi.</note>
            ad verticem, erit quoque CLG, eidem angulo PLN, æqualis.) </s>
            <s xml:id="echoid-s16800" xml:space="preserve">erunt quoq; </s>
            <s xml:id="echoid-s16801" xml:space="preserve">re-
              <lb/>
            liquianguli ALG, ALN, æquales. </s>
            <s xml:id="echoid-s16802" xml:space="preserve">Rurſus quia anguli CLK, CLM, oſtenſi ſunt
              <lb/>
              <note symbol="n" position="right" xlink:label="note-387-13" xlink:href="note-387-13a" xml:space="preserve">26. tertij.</note>
            æquales; </s>
            <s xml:id="echoid-s16803" xml:space="preserve"> erunt arcus CK, CM, æquales. </s>
            <s xml:id="echoid-s16804" xml:space="preserve"> Igitur anguli CGK, CGM: </s>
            <s xml:id="echoid-s16805" xml:space="preserve"> Ideoq;</s>
            <s xml:id="echoid-s16806" xml:space="preserve">
              <note symbol="o" position="right" xlink:label="note-387-14" xlink:href="note-387-14a" xml:space="preserve">ſchol. 29.
                <lb/>
              tertij.</note>
            & </s>
            <s xml:id="echoid-s16807" xml:space="preserve">anguli A G O, AGL, ad verticem æquales erunt: </s>
            <s xml:id="echoid-s16808" xml:space="preserve"> Ac proinde arcus etiam AO, AL, æquales erunt: </s>
            <s xml:id="echoid-s16809" xml:space="preserve"> ideoque & </s>
            <s xml:id="echoid-s16810" xml:space="preserve">anguli AKO, AKL, erunt æquales. </s>
            <s xml:id="echoid-s16811" xml:space="preserve">
              <note symbol="p" position="right" xlink:label="note-387-15" xlink:href="note-387-15a" xml:space="preserve">15 primi</note>
            ducta autem AK, ad Q, ſi ex rectis angulis CKA, CKQ, auferantur æquales AKG,
              <lb/>
              <note symbol="q" position="right" xlink:label="note-387-16" xlink:href="note-387-16a" xml:space="preserve">ſchol. 29.
                <lb/>
              tertij.</note>
            FKQ, (Cum enim angulus AKG, angulo AKL, oſtenſus ſit æqualis: </s>
            <s xml:id="echoid-s16812" xml:space="preserve"> hic autem angulo FKQ, ad verticem ſit æqualis; </s>
            <s xml:id="echoid-s16813" xml:space="preserve">erit quoque angulus AKG, angulo FKQ.
              <lb/>
            </s>
            <s xml:id="echoid-s16814" xml:space="preserve">
              <note symbol="r" position="right" xlink:label="note-387-17" xlink:href="note-387-17a" xml:space="preserve">27. tertij.</note>
            æqualis.) </s>
            <s xml:id="echoid-s16815" xml:space="preserve">erunt etiam reliqui anguli CKG, CKF, inter ſe æquales. </s>
            <s xml:id="echoid-s16816" xml:space="preserve">Quæ omnia
              <lb/>
              <note symbol="s" position="right" xlink:label="note-387-18" xlink:href="note-387-18a" xml:space="preserve">15. primi.</note>
            demonſtranda erant.</s>
            <s xml:id="echoid-s16817" xml:space="preserve"/>
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