Clavius, Christoph, Geometria practica

Table of contents

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[191.] II. EX diametro aream circuli vera minorem inueſtigare.
Page: 227 (197)
[192.] III. EX circumferentia aream circuli vera maiorem colligere.
Page: 227 (197)
[193.] IV. EX circumferentia aream circuli vera minorem concludere.
Page: 227 (197)
[194.] DE AREA SEGMENTORVM CIRCVLI. Capvt VIII.
Page: 229 (199)
[195.] I.
Page: 231 (201)
[196.] II.
Page: 231 (201)
[197.] III.
Page: 231 (201)
[198.] IV.
Page: 232 (202)
[199.] V.
Page: 232 (202)
[200.] VI.
Page: 233 (203)
[201.] FINIS LIBRI QVARTI.
Page: 233 (203)
[202.] GEOMETRIÆ PRACTICÆ LIBER QVINTVS.
Page: 234 (204)
[203.] AREAS Solidorum, corporumue perſcrutans.
Page: 234 (204)
[204.] DE AREA PARALLELEPIP EDO-rum, Priſmatum, & Cylindrorum. Capvt I.
Page: 234 (204)
[205.] DE AREA PYRAMIDVM & Conorum. Capvt II.
Page: 236 (206)
[206.] DL AREA FRVSTI PYRA-midis, & Coni. Capvt III.
Page: 237 (207)
[207.] SCHOLIVM.
Page: 239 (209)
[208.] DE AREA QVINQVE COR-porum regularium. Capvt IV.
Page: 240 (210)
[209.] Capvt V.
Page: 248 (218)
[210.] PROPOSITIO I.
Page: 248 (218)
[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)
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          <p>
            <s xml:id="echoid-s7056" xml:space="preserve">
              <pb o="170" file="200" n="200" rhead="GEOMETR. PRACT."/>
            las AD, BC, & </s>
            <s xml:id="echoid-s7057" xml:space="preserve">ſuper eandẽ baſem AD. </s>
            <s xml:id="echoid-s7058" xml:space="preserve">Itaq; </s>
            <s xml:id="echoid-s7059" xml:space="preserve">fruſtra alij p̃cipiunt, vt diameter du
              <lb/>
            catur AC, & </s>
            <s xml:id="echoid-s7060" xml:space="preserve">beneficio perpendicularis AE, area trianguli ABC, inquiratur, quod
              <lb/>
            hæc duplicata aream exhibeat totius parallelogrammi, quippe cum
              <note symbol="a" position="left" xlink:label="note-200-01" xlink:href="note-200-01a" xml:space="preserve">34. primi.</note>
            lum ABC, ſemiſsis ſit parallelogrammi. </s>
            <s xml:id="echoid-s7061" xml:space="preserve">Fruſtra, inquam hoc præcipiunt, cum
              <lb/>
            expeditius area inueniatur ſi perpendicularis in totum latus BC, ducatur, quam
              <lb/>
            ſi in ſemiſſem multiplicetur, ac productus deinde numerus dupletur.</s>
            <s xml:id="echoid-s7062" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7063" xml:space="preserve">
              <emph style="sc">Si</emph>
            per quadrantem cap. </s>
            <s xml:id="echoid-s7064" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7065" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7066" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7067" xml:space="preserve">conſtructum inueſtigetur quantitas anguli
              <lb/>
              <note position="left" xlink:label="note-200-02" xlink:href="note-200-02a" xml:space="preserve">Perpendicu-
                <lb/>
              laris inuentio.</note>
            B, reperietur perpendicularis AE, perſinus, hacratione. </s>
            <s xml:id="echoid-s7068" xml:space="preserve">Fiat vt ſinus totus an-
              <lb/>
            gulirecti E, ad latus oppoſitum AB, ita ſinus anguli B, ad aliud. </s>
            <s xml:id="echoid-s7069" xml:space="preserve"> Productus
              <note symbol="b" position="left" xlink:label="note-200-03" xlink:href="note-200-03a" xml:space="preserve">2. triang. re-
                <lb/>
              ctil.</note>
            nim numerus erit perpendicularis AE, cognita in partibus lateris dati AB.</s>
            <s xml:id="echoid-s7070" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7071" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7072" xml:space="preserve">
              <emph style="sc">Trapezii</emph>
            , in quo duo latera oppoſita ſint parallela AB, BC, & </s>
            <s xml:id="echoid-s7073" xml:space="preserve">omnia
              <lb/>
            latera nota, area producitur ex perpendiculari A E, inter duo latera parallela
              <lb/>
            multiplicata in ſemiſſem ſummæ ex lateribus parallelis conflatæ. </s>
            <s xml:id="echoid-s7074" xml:space="preserve">Nam ducta
              <lb/>
              <note position="left" xlink:label="note-200-04" xlink:href="note-200-04a" xml:space="preserve">Areatrapezii
                <lb/>
              habentis duo
                <lb/>
              lateraparalle-
                <lb/>
              la.</note>
            diametro AC, area trianguli ABC, producitur ex perpendiculari A E, in ſemiſ-
              <lb/>
            ſem baſis BC, vt cap. </s>
            <s xml:id="echoid-s7075" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7076" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s7077" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7078" xml:space="preserve">dictum eſt: </s>
            <s xml:id="echoid-s7079" xml:space="preserve">Item area trian-
              <lb/>
              <figure xlink:label="fig-200-01" xlink:href="fig-200-01a" number="128">
                <image file="200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/200-01"/>
              </figure>
            guli ACD, ex eadem perpendiculari A E, in ſemiſſem baſis
              <lb/>
            AD: </s>
            <s xml:id="echoid-s7080" xml:space="preserve">Acproinde hæ duæ areæ ſimul aream totius Trapezij A-
              <lb/>
            BCD, conficient. </s>
            <s xml:id="echoid-s7081" xml:space="preserve"> Cum igitur idem fiat ex AE, in ſummam
              <note symbol="c" position="left" xlink:label="note-200-05" xlink:href="note-200-05a" xml:space="preserve">1. ſecundi.</note>
            ſemiſſe rectæ B C, & </s>
            <s xml:id="echoid-s7082" xml:space="preserve">ex ſemiſſe rectæ A D, conflatam, id eſt, in
              <lb/>
            ſemiſſem rectarum BC, AD, ſimul: </s>
            <s xml:id="echoid-s7083" xml:space="preserve">quod ex A E, in ſemiſſem
              <lb/>
            lateris B C, & </s>
            <s xml:id="echoid-s7084" xml:space="preserve">ex A E, in ſemiſſem lateris A D; </s>
            <s xml:id="echoid-s7085" xml:space="preserve">liquidò conſtat,
              <lb/>
            aream Trapezij gigni ex perpendiculari AE, in ſemiſſem ſum-
              <lb/>
            mæ laterum AD, BC. </s>
            <s xml:id="echoid-s7086" xml:space="preserve">Atque hæc ratio locum etiam habetin
              <lb/>
            Trapezio habente vnum angulum rectum, vel duos rectos.</s>
            <s xml:id="echoid-s7087" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7088" xml:space="preserve">
              <emph style="sc">Perpendicvlaris</emph>
            vero AE, inuenietur, vt in Rhombo, & </s>
            <s xml:id="echoid-s7089" xml:space="preserve">Rhomboi-
              <lb/>
            de diximus, duobus modis, ſi per quadrantem angulus B, inueſtigetur, &</s>
            <s xml:id="echoid-s7090" xml:space="preserve">c.</s>
            <s xml:id="echoid-s7091" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7092" xml:space="preserve">
              <emph style="sc">In</emph>
            Trapezio autem FGHI, in quo nulla ſunt latera parallela, omnia tamen
              <lb/>
              <note position="left" xlink:label="note-200-06" xlink:href="note-200-06a" xml:space="preserve">Areatrapezii
                <lb/>
              nulla haben-
                <lb/>
              tis latera pa-
                <lb/>
              rallela.</note>
            latera ſunt nota, menſuranda primum eſt diameter. </s>
            <s xml:id="echoid-s7093" xml:space="preserve">IG, per inſtrumentum par-
              <lb/>
            tium. </s>
            <s xml:id="echoid-s7094" xml:space="preserve">Deinde vtriuſque trianguli FGI, GHI, area inuenienda, vt cap. </s>
            <s xml:id="echoid-s7095" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7096" xml:space="preserve">Nume.
              <lb/>
            </s>
            <s xml:id="echoid-s7097" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7098" xml:space="preserve">& </s>
            <s xml:id="echoid-s7099" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7100" xml:space="preserve">tradidimus. </s>
            <s xml:id="echoid-s7101" xml:space="preserve">Ambæ enim areæ ſimul conficient aream totius Trapezij.</s>
            <s xml:id="echoid-s7102" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7103" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            ſi malueri angulum F, vel H, per quadrantem inuenire, cognoſce-
              <lb/>
            mus diametri GI, magnitudinem, per doctrinam ſinuum, ac Tangentium,
              <lb/>
            vt lib. </s>
            <s xml:id="echoid-s7104" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7105" xml:space="preserve">capit. </s>
            <s xml:id="echoid-s7106" xml:space="preserve">3. </s>
            <s xml:id="echoid-s7107" xml:space="preserve">docuimus, ex duobus lateribus F G, F I, & </s>
            <s xml:id="echoid-s7108" xml:space="preserve">angulo F,
              <note symbol="d" position="left" xlink:label="note-200-07" xlink:href="note-200-07a" xml:space="preserve">12. trian. re-
                <lb/>
              ctil. Num. 2.</note>
            ipſis comprehenſo, vel ex duobus lateribus HG, HI, & </s>
            <s xml:id="echoid-s7109" xml:space="preserve">angulo H, quem con-
              <lb/>
            tinent.</s>
            <s xml:id="echoid-s7110" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7111" xml:space="preserve">3. </s>
            <s xml:id="echoid-s7112" xml:space="preserve">
              <emph style="sc">Non</emph>
            aliter aream conſequemur cuiuſcun que quadrilateri irregularis, et-
              <lb/>
              <note position="left" xlink:label="note-200-08" xlink:href="note-200-08a" xml:space="preserve">Area figuræ
                <lb/>
              quadrilateræ
                <lb/>
              irregularis.</note>
            iamſi non habeat omnes angulos introrſum, ſicut Trapezium. </s>
            <s xml:id="echoid-s7113" xml:space="preserve">Vt ſi in Trape-
              <lb/>
            zio FGHI, ducantur ex G, & </s>
            <s xml:id="echoid-s7114" xml:space="preserve">I, duæ rectæ GK, IK, conſtituetur quadrilaterum
              <lb/>
            GHIK, irregulare, cum ſolum habeat tres angulos GHI, HIK, HGK. </s>
            <s xml:id="echoid-s7115" xml:space="preserve">Nam ad
              <lb/>
            K, non fit angulus GKI, introrſum verſus H, cum illud ſpatium ſit duo-
              <lb/>
            bus rectis maius, ſed verſus F, extrorſum. </s>
            <s xml:id="echoid-s7116" xml:space="preserve">Huius ergo figuræ qua-
              <lb/>
            drilateræirregularis aream colligemus, ducta diametro
              <lb/>
            K H, ex duabus areis triangulorum IKH,
              <lb/>
            GKH, vt de Trapezio FGHI,
              <lb/>
            dictum eſt.</s>
            <s xml:id="echoid-s7117" xml:space="preserve"/>
          </p>
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