Clavius, Christoph, Geometria practica

Table of contents

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[211.] COROLLARIVM.
Page: 249 (219)
[212.] PROPOSITIO II.
Page: 249 (219)
[213.] COROLLARIVM.
Page: 249 (219)
[214.] PROPOSITIO III.
Page: 250 (220)
[215.] COROLLARIVM.
Page: 250 (220)
[216.] PROPOSITIO IV.
Page: 251 (221)
[217.] PROPOSITIO V.
Page: 251 (221)
[218.] PROPOSITIO VI.
Page: 251 (221)
[219.] PROPOSITIO VII.
Page: 252 (222)
[220.] I.
Page: 253 (223)
[221.] II.
Page: 253 (223)
[222.] ALITER.
Page: 255 (225)
[223.] ALITER.
Page: 255 (225)
[224.] I.
Page: 256 (226)
[225.] II.
Page: 257 (227)
[226.] III.
Page: 257 (227)
[227.] IIII.
Page: 257 (227)
[228.] I.
Page: 258 (228)
[229.] II.
Page: 258 (228)
[230.] III.
Page: 258 (228)
[231.] IIII.
Page: 258 (228)
[232.] DE AREA SEGMENTO-rum ſphæræ. Capvt VI.
Page: 259 (229)
[233.] ALITER.
Page: 261 (231)
[234.] DE AREA SPHÆROIDIS, EIVSDEM-que portionum. Capvt VII.
Page: 262 (232)
[235.] DE AREA CONOIDIS parabolici. Capvt VIII.
Page: 262 (232)
[236.] DE AREA CONOIDIS Hyperbolici. Capvt IX.
Page: 263 (233)
[237.] DE AREA DOLIORVM. Capvt X.
Page: 263 (233)
[238.] DE AREA CORPORVM. omnino irregularium. Capvt XI.
Page: 264 (234)
[239.] DE SVPERFICIE CONVEXA coni & cylindri recti. Capvt XII.
Page: 265 (235)
[240.] FINIS LIBRI QVINTI.
Page: 265 (235)
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          <p>
            <s xml:id="echoid-s8074" xml:space="preserve">
              <pb o="188" file="218" n="218" rhead="GEOMETR. PRACT."/>
            latus erit polygoni circulo circumſcripti, quod lateribus æqualibus 96. </s>
            <s xml:id="echoid-s8075" xml:space="preserve">conti-
              <lb/>
            netur. </s>
            <s xml:id="echoid-s8076" xml:space="preserve">Et quia oſtenſum eſt, E c, ad c o, maiorem habere proportionem, quam
              <lb/>
            4673 {1/2}. </s>
            <s xml:id="echoid-s8077" xml:space="preserve">ad 153. </s>
            <s xml:id="echoid-s8078" xml:space="preserve"> habebit quo que diameter AB, ipſius E c, dupla ad o i, ipſius c
              <note symbol="a" position="left" xlink:label="note-218-01" xlink:href="note-218-01a" xml:space="preserve">15. quinti.</note>
            duplam maiorem proportionem, quam 4673 {1/2}. </s>
            <s xml:id="echoid-s8079" xml:space="preserve">ad 153. </s>
            <s xml:id="echoid-s8080" xml:space="preserve">Si ergo o i, latus polygo-
              <lb/>
            ni ponatur 153. </s>
            <s xml:id="echoid-s8081" xml:space="preserve"> erit diameter AB, maior, quam 4673 {1/2}. </s>
            <s xml:id="echoid-s8082" xml:space="preserve">Multiplicentur 153. </s>
            <s xml:id="echoid-s8083" xml:space="preserve">
              <note symbol="b" position="left" xlink:label="note-218-02" xlink:href="note-218-02a" xml:space="preserve">10. quinti.</note>
            96. </s>
            <s xml:id="echoid-s8084" xml:space="preserve">vt totus ambitus polygoni producatur 14688. </s>
            <s xml:id="echoid-s8085" xml:space="preserve"> habebitque ambitus
              <note symbol="c" position="left" xlink:label="note-218-03" xlink:href="note-218-03a" xml:space="preserve">8. quinti.</note>
            goni ad diametrum AB, minorem proportionem, quam 14688. </s>
            <s xml:id="echoid-s8086" xml:space="preserve">ad 4673 {1/2}. </s>
            <s xml:id="echoid-s8087" xml:space="preserve">
              <note symbol="d" position="left" xlink:label="note-218-04" xlink:href="note-218-04a" xml:space="preserve">8. quinti.</note>
            autem proportio 14688. </s>
            <s xml:id="echoid-s8088" xml:space="preserve">ad 4673 {1/2}. </s>
            <s xml:id="echoid-s8089" xml:space="preserve">minor, quam tripla ſeſquiſeptima; </s>
            <s xml:id="echoid-s8090" xml:space="preserve">quod
              <lb/>
            14688. </s>
            <s xml:id="echoid-s8091" xml:space="preserve">ad 4673. </s>
            <s xml:id="echoid-s8092" xml:space="preserve">{5/11}. </s>
            <s xml:id="echoid-s8093" xml:space="preserve">(quinumerus paulo minor eſt quam 4673 {1/2}.) </s>
            <s xml:id="echoid-s8094" xml:space="preserve">habeant pro-
              <lb/>
            portionem triplam ſeſquiſeptimam. </s>
            <s xml:id="echoid-s8095" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s8096" xml:space="preserve">ambitus polygoni ad
              <note symbol="e" position="left" xlink:label="note-218-05" xlink:href="note-218-05a" xml:space="preserve">ſchol. 13.
                <lb/>
              quinti.</note>
            AB, proportionem habet minorem tripla ſeſquiſeptima: </s>
            <s xml:id="echoid-s8097" xml:space="preserve"> atque adeo circum- ferentia, quæ (vt lib. </s>
            <s xml:id="echoid-s8098" xml:space="preserve">8. </s>
            <s xml:id="echoid-s8099" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s8100" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8101" xml:space="preserve">probabimus) minor eſt ambitu polygoni, mul-
              <lb/>
              <note symbol="f" position="left" xlink:label="note-218-06" xlink:href="note-218-06a" xml:space="preserve">8 quinti.</note>
            to minorem proportionem tripla ſeſquiſeptima ad diametrum habebit; </s>
            <s xml:id="echoid-s8102" xml:space="preserve">ideoq;
              <lb/>
            </s>
            <s xml:id="echoid-s8103" xml:space="preserve">circumferentia tripla eſt diametri, & </s>
            <s xml:id="echoid-s8104" xml:space="preserve">ad huc ſuperat parte, quæ minor eſt {30/70}. </s>
            <s xml:id="echoid-s8105" xml:space="preserve">hoc
              <lb/>
            eſt, {1/7}. </s>
            <s xml:id="echoid-s8106" xml:space="preserve">diametri. </s>
            <s xml:id="echoid-s8107" xml:space="preserve">Nam ſi contineret ter, & </s>
            <s xml:id="echoid-s8108" xml:space="preserve">{1/7}. </s>
            <s xml:id="echoid-s8109" xml:space="preserve">haberet cir cumferentia ad diame-
              <lb/>
            trum proportionem triplam ſeſquiſeptimam: </s>
            <s xml:id="echoid-s8110" xml:space="preserve">ſi vero contineret ter, & </s>
            <s xml:id="echoid-s8111" xml:space="preserve">pluſquã
              <lb/>
            {1/7}. </s>
            <s xml:id="echoid-s8112" xml:space="preserve">haberet maiorem proportionem, quam triplam ſeſquiſep timam, cum tamen
              <lb/>
            minorem habeat, vt demonſtratum eſt.</s>
            <s xml:id="echoid-s8113" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8114" xml:space="preserve">
              <emph style="sc">Iam</emph>
            vero in eodem circulo ſit latus hexagoni B G, ſemidiametro æquale,
              <lb/>
            per coroll. </s>
            <s xml:id="echoid-s8115" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s8116" xml:space="preserve">15. </s>
            <s xml:id="echoid-s8117" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s8118" xml:space="preserve">4. </s>
            <s xml:id="echoid-s8119" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s8120" xml:space="preserve">iunganturque rectæ A G, E G. </s>
            <s xml:id="echoid-s8121" xml:space="preserve">Et quia trian-
              <lb/>
            gulum E B G, eſt æquilaterum conſtans ex tribus ſemidiametris; </s>
            <s xml:id="echoid-s8122" xml:space="preserve"> erit
              <note symbol="g" position="left" xlink:label="note-218-07" xlink:href="note-218-07a" xml:space="preserve">coroll. 3 pro-
                <lb/>
              poſ. 32. lib. 1.</note>
            B E G, {2/3}. </s>
            <s xml:id="echoid-s8123" xml:space="preserve">vnius recti, ac proinde eius ſemiſsis B A G, erit {1/3}. </s>
            <s xml:id="echoid-s8124" xml:space="preserve">vnius recti. </s>
            <s xml:id="echoid-s8125" xml:space="preserve">Et quia diameter AB, dupla eſt ſemidiametri BG, ſi BG, ponatur 780. </s>
            <s xml:id="echoid-s8126" xml:space="preserve">erit AB, 1560. </s>
            <s xml:id="echoid-s8127" xml:space="preserve">
              <note symbol="h" position="left" xlink:label="note-218-08" xlink:href="note-218-08a" xml:space="preserve">20. tertij.</note>
            ergo quadratumipſius A B, æquale ſit quadratis rectarum B G, G A; </s>
            <s xml:id="echoid-s8128" xml:space="preserve"> quod
              <note symbol="i" position="left" xlink:label="note-218-09" xlink:href="note-218-09a" xml:space="preserve">47. primi.</note>
            gulus AGB, in ſemicirculo rectus ſit: </s>
            <s xml:id="echoid-s8129" xml:space="preserve">ſi quadratum 608400. </s>
            <s xml:id="echoid-s8130" xml:space="preserve">ipſius B G, dema-
              <lb/>
              <note symbol="k" position="left" xlink:label="note-218-10" xlink:href="note-218-10a" xml:space="preserve">31. tertij.</note>
            tur ex 2433600. </s>
            <s xml:id="echoid-s8131" xml:space="preserve">quadrato ipſius A B, reliquum fiet quadratum 1825200. </s>
            <s xml:id="echoid-s8132" xml:space="preserve">ipſius
              <lb/>
            AG, cuius radix paulo minor eſt, quam 1351. </s>
            <s xml:id="echoid-s8133" xml:space="preserve">cum huius quadratum 1825201.
              <lb/>
            </s>
            <s xml:id="echoid-s8134" xml:space="preserve">maius ſit, quam 1825200. </s>
            <s xml:id="echoid-s8135" xml:space="preserve"> Igitur AG, ad GB, minorem habebit
              <note symbol="l" position="left" xlink:label="note-218-11" xlink:href="note-218-11a" xml:space="preserve">8. quinti.</note>
            quam 1351. </s>
            <s xml:id="echoid-s8136" xml:space="preserve">ad 780. </s>
            <s xml:id="echoid-s8137" xml:space="preserve">ac proinde ſi B G, ponatur 780. </s>
            <s xml:id="echoid-s8138" xml:space="preserve"> erit A G, minor, quam 1351.</s>
            <s xml:id="echoid-s8139" xml:space="preserve"/>
          </p>
          <note symbol="m" position="left" xml:space="preserve">10. quinti.</note>
          <p>
            <s xml:id="echoid-s8140" xml:space="preserve">
              <emph style="sc">Secto</emph>
            iam angulo BAG, bifariam per rectam AH, ſecantem BG, in M, du-
              <lb/>
            ctaque HB, erunt triangula BHM, AHB, æquiangula: </s>
            <s xml:id="echoid-s8141" xml:space="preserve"> propterea quod
              <note symbol="n" position="left" xlink:label="note-218-13" xlink:href="note-218-13a" xml:space="preserve">21. tertij.</note>
            lus HBM, æqualis eſt angulo HAG, ob eandem baſem GH; </s>
            <s xml:id="echoid-s8142" xml:space="preserve">ideoque per con-
              <lb/>
            ſtru ctionem angulo HAB; </s>
            <s xml:id="echoid-s8143" xml:space="preserve">& </s>
            <s xml:id="echoid-s8144" xml:space="preserve">angulus rectus H, in ſemicirculo communis. </s>
            <s xml:id="echoid-s8145" xml:space="preserve">
              <note symbol="o" position="left" xlink:label="note-218-14" xlink:href="note-218-14a" xml:space="preserve">4. ſexti.</note>
            gitur erit AH, ad HB, vt HB, ad HM. </s>
            <s xml:id="echoid-s8146" xml:space="preserve">Item AB, ad BH, vt BM, ad MH: </s>
            <s xml:id="echoid-s8147" xml:space="preserve">& </s>
            <s xml:id="echoid-s8148" xml:space="preserve">per-
              <lb/>
            mutando AB, ad BM, vt BH, ad HM: </s>
            <s xml:id="echoid-s8149" xml:space="preserve">ideoque erunt tres hæ proportiones AH,
              <lb/>
            ad HB; </s>
            <s xml:id="echoid-s8150" xml:space="preserve">HB, ad HM: </s>
            <s xml:id="echoid-s8151" xml:space="preserve">& </s>
            <s xml:id="echoid-s8152" xml:space="preserve">AB, ad BM, æquales. </s>
            <s xml:id="echoid-s8153" xml:space="preserve">Sed vt AB, ad BM, ita eſt vtraq;
              <lb/>
            </s>
            <s xml:id="echoid-s8154" xml:space="preserve">ſimul BA, AG, ad BG. </s>
            <s xml:id="echoid-s8155" xml:space="preserve"> Nam vt AG, ad AB, ita eſt GM, ad MB; </s>
            <s xml:id="echoid-s8156" xml:space="preserve">& </s>
            <s xml:id="echoid-s8157" xml:space="preserve">componendo
              <note symbol="p" position="left" xlink:label="note-218-15" xlink:href="note-218-15a" xml:space="preserve">3. ſexti.</note>
            AG, AB, ſimul ad AB, ita GM, MB, ſimul id eſt, tota G B, ad M B: </s>
            <s xml:id="echoid-s8158" xml:space="preserve">Et permut ando vt
              <lb/>
            A G, A B, ſimulad G B, ita A B, ad M B. </s>
            <s xml:id="echoid-s8159" xml:space="preserve">Igitur erit quoque, vt vtraque AG, AB, ſi-
              <lb/>
            mul ad GB, ita AH, ad HB. </s>
            <s xml:id="echoid-s8160" xml:space="preserve">Eſt autem AG, oſtenſa minor, quam 1351. </s>
            <s xml:id="echoid-s8161" xml:space="preserve">& </s>
            <s xml:id="echoid-s8162" xml:space="preserve">AB, po-
              <lb/>
            ſita eſt 1560. </s>
            <s xml:id="echoid-s8163" xml:space="preserve">& </s>
            <s xml:id="echoid-s8164" xml:space="preserve">GB, 780. </s>
            <s xml:id="echoid-s8165" xml:space="preserve">Igitur vtraque AG, AB, ſimul (cum minus effi ciant, quã
              <lb/>
            2911.) </s>
            <s xml:id="echoid-s8166" xml:space="preserve"> minorem habebit proportionem ad GB, quam 2911. </s>
            <s xml:id="echoid-s8167" xml:space="preserve">ad 780. </s>
            <s xml:id="echoid-s8168" xml:space="preserve">Quare
              <note symbol="q" position="left" xlink:label="note-218-16" xlink:href="note-218-16a" xml:space="preserve">8. quinti.</note>
            iam proportio AH, ad HB, minor erit, quam 2911. </s>
            <s xml:id="echoid-s8169" xml:space="preserve">ad 780. </s>
            <s xml:id="echoid-s8170" xml:space="preserve">ac proinde ſi HB, po-
              <lb/>
            natur 780. </s>
            <s xml:id="echoid-s8171" xml:space="preserve"> erit AH, minor, quam 2911. </s>
            <s xml:id="echoid-s8172" xml:space="preserve">ideoque eius quadratum minus,
              <note symbol="r" position="left" xlink:label="note-218-17" xlink:href="note-218-17a" xml:space="preserve">10. quinti.</note>
            8473921. </s>
            <s xml:id="echoid-s8173" xml:space="preserve">cui ſi addatur quadratum 608400. </s>
            <s xml:id="echoid-s8174" xml:space="preserve">ipſius B H, fiet quadratum ipſius
              <lb/>
            AB, (quod quadratis rectarum AH, HB, æquale eſt) minus, quam 9082321. </s>
            <s xml:id="echoid-s8175" xml:space="preserve">
              <note symbol="s" position="left" xlink:label="note-218-18" xlink:href="note-218-18a" xml:space="preserve">47. primi.</note>
            co que eius radix, vel recta A B, minor, quam 3013 {3/4}. </s>
            <s xml:id="echoid-s8176" xml:space="preserve">cum huius </s>
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