Clavius, Christoph, Geometria practica

Table of contents

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[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)
[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)
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          <pb o="178" file="208" n="208" rhead="GEOMETR. PRACT."/>
          <p>
            <s xml:id="echoid-s7453" xml:space="preserve"> inſuper Secunda, Tertia, Quarta, &</s>
            <s xml:id="echoid-s7454" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7455" xml:space="preserve">vt earum areas conſequi poſsimus. </s>
            <s xml:id="echoid-s7456" xml:space="preserve">Nam
              <lb/>
            ex latere cuiuſcunque figurę regularis cognito in partibus diametri circuli cir-
              <lb/>
            cumſcripti, vel ſinus totius, veniemus per ea, quæ cap. </s>
            <s xml:id="echoid-s7457" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7458" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s7459" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7460" xml:space="preserve">ſcripſimus, in
              <lb/>
            cognitionem lineæ perpendicularis ex centro in vnum latus deductę, ac proin-
              <lb/>
            de totam aream nanciſcemur, vt paulo ante Num. </s>
            <s xml:id="echoid-s7461" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7462" xml:space="preserve">docuimus. </s>
            <s xml:id="echoid-s7463" xml:space="preserve">Latera igitur in
              <lb/>
            quam plurimis figuris in tabula pręcedenti expoſita habes, quæ omnia veris la-
              <lb/>
            teribus ſunt paulò minora; </s>
            <s xml:id="echoid-s7464" xml:space="preserve">& </s>
            <s xml:id="echoid-s7465" xml:space="preserve">ſi adieceris vnitatem, fient paulò maiora veris;
              <lb/>
            </s>
            <s xml:id="echoid-s7466" xml:space="preserve">ita vt verum latus trianguli æquilateri inter hos duos numeros 17320508. </s>
            <s xml:id="echoid-s7467" xml:space="preserve">
              <lb/>
            17320509. </s>
            <s xml:id="echoid-s7468" xml:space="preserve">conſiſtat.</s>
            <s xml:id="echoid-s7469" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Ex cognita ſe-
            <lb/>
          midiametro
            <lb/>
          circuli inue-
            <lb/>
          nire lat{us} fi-
            <lb/>
          guræregula-
            <lb/>
          ris in eo circu-
            <lb/>
          lo deſcriptæ.</note>
          <p>
            <s xml:id="echoid-s7470" xml:space="preserve">4. </s>
            <s xml:id="echoid-s7471" xml:space="preserve">
              <emph style="sc">Iam</emph>
            verò cognita ſemidiametro alicuius circuli in partibus cuiuſcunque
              <lb/>
            menſuræ, reperiemus in iiſdem partibus latus figurę regularis, cuius laterum nu-
              <lb/>
            merus maior non eſt, quam 80. </s>
            <s xml:id="echoid-s7472" xml:space="preserve">beneficio præcedentis tabulę: </s>
            <s xml:id="echoid-s7473" xml:space="preserve">ſi nimirum fiat,
              <lb/>
            vt ſinus totus 10000000. </s>
            <s xml:id="echoid-s7474" xml:space="preserve">ad latus figurę propoſitæ in præcedenti tabula, ita ſe-
              <lb/>
            midiamer circuli propoſiti data ad aliud. </s>
            <s xml:id="echoid-s7475" xml:space="preserve">Sit verbi gratia, ſemidiameter alicuius
              <lb/>
            circuli 12. </s>
            <s xml:id="echoid-s7476" xml:space="preserve">& </s>
            <s xml:id="echoid-s7477" xml:space="preserve">inueniendum ſit latus decagoni reſpectu dictæ ſemidiametri: </s>
            <s xml:id="echoid-s7478" xml:space="preserve">Fiat
              <lb/>
            vt 10000000. </s>
            <s xml:id="echoid-s7479" xml:space="preserve">ſinus totus ad 6180339. </s>
            <s xml:id="echoid-s7480" xml:space="preserve">latus Decagoni; </s>
            <s xml:id="echoid-s7481" xml:space="preserve">ita 12. </s>
            <s xml:id="echoid-s7482" xml:space="preserve">ſemidiameter da-
              <lb/>
            ta ad aliud; </s>
            <s xml:id="echoid-s7483" xml:space="preserve">exibitque latus quæſitum 8 {164068/10000000}. </s>
            <s xml:id="echoid-s7484" xml:space="preserve">vel in minoribus numeris
              <lb/>
            8 {41017/2500000}.</s>
            <s xml:id="echoid-s7485" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Fractionem
            <lb/>
          magnam ad
            <lb/>
          minorem ferè
            <lb/>
          æquiualentem
            <lb/>
          reducere.</note>
          <p>
            <s xml:id="echoid-s7486" xml:space="preserve">
              <emph style="sc">Et</emph>
            ſi moleſtum videatur operari cum fractione tam magna, reduces eam ad
              <lb/>
            minorem quaſi æquiualentem hoc modo. </s>
            <s xml:id="echoid-s7487" xml:space="preserve">Elige pro Numeratore quemuis nu-
              <lb/>
            merum, vt 10. </s>
            <s xml:id="echoid-s7488" xml:space="preserve">Et fiat vt Numerator 164068. </s>
            <s xml:id="echoid-s7489" xml:space="preserve">ad ſuum Denominatorẽ 10000000.
              <lb/>
            </s>
            <s xml:id="echoid-s7490" xml:space="preserve">ita Numerator electus 10. </s>
            <s xml:id="echoid-s7491" xml:space="preserve">ad aliud, reperieſque Denominatorem 609 {82588/164068}. </s>
            <s xml:id="echoid-s7492" xml:space="preserve">
              <lb/>
            Ita vt relicta hac fractione, Denominator 609. </s>
            <s xml:id="echoid-s7493" xml:space="preserve">ſit minor quam verus; </s>
            <s xml:id="echoid-s7494" xml:space="preserve">& </s>
            <s xml:id="echoid-s7495" xml:space="preserve">610. </s>
            <s xml:id="echoid-s7496" xml:space="preserve">
              <lb/>
            maior, hoc eſt, fractio {10/609}. </s>
            <s xml:id="echoid-s7497" xml:space="preserve">ſit maior fractione {164068/10000000}. </s>
            <s xml:id="echoid-s7498" xml:space="preserve">fractio autem
              <lb/>
              <note position="left" xlink:label="note-208-03" xlink:href="note-208-03a" xml:space="preserve">Inter du{as}
                <lb/>
              fractiones in-
                <lb/>
              uenire mediã.</note>
            {10/610}. </s>
            <s xml:id="echoid-s7499" xml:space="preserve">minor Inter has autẽ duas fractiones {10/609}. </s>
            <s xml:id="echoid-s7500" xml:space="preserve">{10/610}. </s>
            <s xml:id="echoid-s7501" xml:space="preserve">produces mediã {20/1219}. </s>
            <s xml:id="echoid-s7502" xml:space="preserve">cuius
              <lb/>
            Numerator ex Numeratoribus, & </s>
            <s xml:id="echoid-s7503" xml:space="preserve">Denominator ex Denominatoribus confla-
              <lb/>
            tus eſt. </s>
            <s xml:id="echoid-s7504" xml:space="preserve">Erit que fractio inuenta ferè maiori illi æqualis.</s>
            <s xml:id="echoid-s7505" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7506" xml:space="preserve">
              <emph style="sc">Fractionem</emph>
            porrò, cuius Numerator ex duobus Numeratoribus, & </s>
            <s xml:id="echoid-s7507" xml:space="preserve">De-
              <lb/>
            nominator ex Denominatoribus duarum minutiarum componitur, eſſe maio-
              <lb/>
            rem minore, & </s>
            <s xml:id="echoid-s7508" xml:space="preserve">minorem maiore, demonſtrabimus lib. </s>
            <s xml:id="echoid-s7509" xml:space="preserve">8. </s>
            <s xml:id="echoid-s7510" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s7511" xml:space="preserve">10.</s>
            <s xml:id="echoid-s7512" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Ex cognito
            <lb/>
          latere figuræ
            <lb/>
          regularis, in-
            <lb/>
          uenire ſemi-
            <lb/>
          diam{et}rum
            <lb/>
          circuli cir-
            <lb/>
          cumſcripti.</note>
          <p>
            <s xml:id="echoid-s7513" xml:space="preserve">
              <emph style="sc">Vicissim</emph>
            ex dato latere cuiuslibet figuræ regularis cognoſcemus ſemi-
              <lb/>
            diametrum circuli circumſcribentis: </s>
            <s xml:id="echoid-s7514" xml:space="preserve">ſi fiat, vt latus propoſitæ figuræ in tabula
              <lb/>
            antecedente ad 10000000. </s>
            <s xml:id="echoid-s7515" xml:space="preserve">ita latus datum ad aliud. </s>
            <s xml:id="echoid-s7516" xml:space="preserve">Vt ſi latus Pentagoni de-
              <lb/>
            tur 12. </s>
            <s xml:id="echoid-s7517" xml:space="preserve">& </s>
            <s xml:id="echoid-s7518" xml:space="preserve">fiat, vt 11755705. </s>
            <s xml:id="echoid-s7519" xml:space="preserve">ad 10000000. </s>
            <s xml:id="echoid-s7520" xml:space="preserve">ita 12. </s>
            <s xml:id="echoid-s7521" xml:space="preserve">ad aliud, reperietur ſemidiame-
              <lb/>
            ter circuli circumſcripti 10 {2442950/11755705}. </s>
            <s xml:id="echoid-s7522" xml:space="preserve">Et ſi fiat, vt Numerator huius fractionis ad
              <lb/>
            ſuum Denominatorem: </s>
            <s xml:id="echoid-s7523" xml:space="preserve">Ita Numerator electus quicunque, nimirum 1. </s>
            <s xml:id="echoid-s7524" xml:space="preserve">ad a-
              <lb/>
            liud, inuenietur Denominator ſequens 4 {1983905/2442950}. </s>
            <s xml:id="echoid-s7525" xml:space="preserve">ita vt fractio {1/4}. </s>
            <s xml:id="echoid-s7526" xml:space="preserve">ſit maior,
              <lb/>
            quã {2442950/11755705}. </s>
            <s xml:id="echoid-s7527" xml:space="preserve">at {1/5}. </s>
            <s xml:id="echoid-s7528" xml:space="preserve">minor Ex additione numeratorũ 1. </s>
            <s xml:id="echoid-s7529" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7530" xml:space="preserve">inter ſe, & </s>
            <s xml:id="echoid-s7531" xml:space="preserve">Denomina-
              <lb/>
            rum 4. </s>
            <s xml:id="echoid-s7532" xml:space="preserve">5. </s>
            <s xml:id="echoid-s7533" xml:space="preserve">inter ſe, efficies fractionem {2/9}. </s>
            <s xml:id="echoid-s7534" xml:space="preserve">mediam, quæ adhuc maior eſt, quam
              <lb/>
            {2442950/11755705}. </s>
            <s xml:id="echoid-s7535" xml:space="preserve">Media autem inter {1/5}. </s>
            <s xml:id="echoid-s7536" xml:space="preserve">& </s>
            <s xml:id="echoid-s7537" xml:space="preserve">{2/9}. </s>
            <s xml:id="echoid-s7538" xml:space="preserve">eſt {3/14}. </s>
            <s xml:id="echoid-s7539" xml:space="preserve">quæ parum ab illa differt: </s>
            <s xml:id="echoid-s7540" xml:space="preserve">ita vt
              <lb/>
            ſemidiameter quæſita dici poſsit eſſe 10 {3/14}. </s>
            <s xml:id="echoid-s7541" xml:space="preserve">Atque in hunc modum per mino-
              <lb/>
            res numeros operationes fieri poſſunt, quamuis non omnino ex quiſitè, quod
              <lb/>
            fractiones aſſumptę non ſint omnino veræ; </s>
            <s xml:id="echoid-s7542" xml:space="preserve">ſed hic error in dimenſionibus cam-
              <lb/>
            porum tolerabilis eſt.</s>
            <s xml:id="echoid-s7543" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7544" xml:space="preserve">5. </s>
            <s xml:id="echoid-s7545" xml:space="preserve">
              <emph style="sc">Anteqvam</emph>
            rectilinearum figurarum dimenſionem concludam, lubet
              <lb/>
            regulam attexere, qua ex cognita area cuiuſcunque figuræ latus habentis </s>
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