Clavius, Christoph, Geometria practica

Table of contents

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[161.] ALITER.
[162.] PROBLEMA XLI.
[163.] PROBLEMA XLII.
[164.] PROBLEMA XLIII.
[165.] PROBLEMA XLIV.
[166.] SCHOLIVM.
[167.] PROBLEMA XLV.
[168.] FINIS LIBRI TERTII.
[169.] GEOMETRIÆ PRACTICÆ LIBER QVARTVS.
[170.] AREAS
[171.] DE AREA RECTANGVLORVM Capvt I.
[172.] DE AREA TRIANGVLORVM Capvt II.
[173.] DE AREA QVADRILATERORVM non rectangulorum. Capvt III.
[174.] DE AREA MVLTIL ATERARVM figurarum irregularium. Capvt IV.
[175.] DE AREA MVLTILATERA-rum figurarum regularium. Capvt V.
[176.] De dimenſione circuli ex Archimede. Capvt VI.
[177.] PROPOSITIO I.
[178.] SCHOLIVM.
[179.] PROPOSITIO II.
[180.] COROLLARIVM.
[181.] PROPOSITIO III.
[182.] DE AREA CIRCVLI, INVENTIONE-que circumferentiæ ex diametro, & diametri ex circumfetentia. Capvt VII.
[184.] II.
[185.] III.
[186.] IIII.
[187.] PROPOSITIO I.
[188.] PROPOSITIO II.
[189.] PROPOSITIO III.
[190.] I. EX diametro aream circuli vera maiorem inueſtigare.
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          <p>
            <s xml:id="echoid-s3953" xml:space="preserve">
              <pb o="102" file="132" n="132" rhead="GEOMETR. PRACT."/>
            totam ita BI, ablata ipſi b H, æqualis, ad ablatum a F. </s>
            <s xml:id="echoid-s3954" xml:space="preserve"> Igitur erit & </s>
            <s xml:id="echoid-s3955" xml:space="preserve">reliqua I
              <note symbol="a" position="left" xlink:label="note-132-01" xlink:href="note-132-01a" xml:space="preserve">19. quinti.</note>
            ad reliquam A a, vt tota B E, ad totam AF. </s>
            <s xml:id="echoid-s3956" xml:space="preserve">Quapropter ſi fiat.</s>
            <s xml:id="echoid-s3957" xml:space="preserve"/>
          </p>
          <note style="it" position="right" xml:space="preserve">
            <lb/>
          Vt I E, differen- \\ tia vmbrarum \\ rectarum # ad A a, differen- \\ tiam ſtationum: # Ita B E, vmbra recta re- \\ motioris ſtationis, ſiue \\ maior # ad AF, di- \\ ſtantiam,
            <lb/>
          </note>
          <p>
            <s xml:id="echoid-s3958" xml:space="preserve">procreabitur AF, diſtantia nota in partibus differentiæ ſtationum A a, notæ.</s>
            <s xml:id="echoid-s3959" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3960" xml:space="preserve">4. </s>
            <s xml:id="echoid-s3961" xml:space="preserve">
              <emph style="sc">Eadem</emph>
            omnino in quadrato pendulo eſt ratio. </s>
            <s xml:id="echoid-s3962" xml:space="preserve">Nam filum perpendi-
              <lb/>
            culi ab ſcindit quo que triangula ABE, ab H, triangulis A F G, aFG, æquiangula;
              <lb/>
            </s>
            <s xml:id="echoid-s3963" xml:space="preserve">
              <note symbol="b" position="left" xlink:label="note-132-03" xlink:href="note-132-03a" xml:space="preserve">29. primi.</note>
            quod tam anguli B, F, recti ſint, & </s>
            <s xml:id="echoid-s3964" xml:space="preserve">angulus BAE angulo AGF, externus inter- no æqualis, quam anguli b, F, recti, & </s>
            <s xml:id="echoid-s3965" xml:space="preserve">angulus b a H, angulo a GF, æqualis, ex-
              <lb/>
            ternus interno. </s>
            <s xml:id="echoid-s3966" xml:space="preserve">Reliqua demonſtrabuntur, vt in ſtabili quadrato. </s>
            <s xml:id="echoid-s3967" xml:space="preserve">Sunt enim
              <lb/>
            vmbræ rectæ in quadrato pendulo vmbris rectis in ſtabili æquales. </s>
            <s xml:id="echoid-s3968" xml:space="preserve">Nam cum
              <lb/>
            duo anguli B, E, in triangulo A B E, quadrati penduli, æquales ſint duobus an-
              <lb/>
            gulis B, E, in triangulo A B E, quadrati ſtabilis; </s>
            <s xml:id="echoid-s3969" xml:space="preserve">quod hæc triangula ſint, vt o-
              <lb/>
            ſtenſum eſt, æquiangula, vt pote æquiangula triangulo A G F; </s>
            <s xml:id="echoid-s3970" xml:space="preserve"> erunt & </s>
            <s xml:id="echoid-s3971" xml:space="preserve">
              <note symbol="c" position="left" xlink:label="note-132-04" xlink:href="note-132-04a" xml:space="preserve">26. primi.</note>
            B E, B E, hoc eſt, vmbræ rectæ æquales.
              <lb/>
            </s>
            <s xml:id="echoid-s3972" xml:space="preserve">Eademque ratione vmbrærectæ b H, b H, æ-
              <lb/>
              <figure xlink:label="fig-132-01" xlink:href="fig-132-01a" number="60">
                <image file="132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/132-01"/>
              </figure>
            quales erunt &</s>
            <s xml:id="echoid-s3973" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3974" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3975" xml:space="preserve">5. </s>
            <s xml:id="echoid-s3976" xml:space="preserve">
              <emph style="sc">Si</emph>
            denique in reniotiore ſtatione ſece-
              <lb/>
              <note position="left" xlink:label="note-132-05" xlink:href="note-132-05a" xml:space="preserve">Ad dexterum
                <lb/>
              angulum ſu-
                <lb/>
              perioremprio-
                <lb/>
              ris quadrati
                <lb/>
              pone C. ad de-
                <lb/>
              xterum infe-
                <lb/>
              riorem poſte-
                <lb/>
              rioris d.</note>
            turvmbra verſa C D, in E, & </s>
            <s xml:id="echoid-s3977" xml:space="preserve">recta b c, in H, in
              <lb/>
            ſtatione propinquiore, reducenda erit alteru-
              <lb/>
            tra earum ad alteram, vt habeantur ſimiles vm-
              <lb/>
            bræ, per ea, quæ in quadrati conſtructione Nu.
              <lb/>
            </s>
            <s xml:id="echoid-s3978" xml:space="preserve">7. </s>
            <s xml:id="echoid-s3979" xml:space="preserve">ad initium huius libritradidimus, diuidendo
              <lb/>
            nimirum quadratum lateris A B, per vmbram,
              <lb/>
            quæreduci debet, &</s>
            <s xml:id="echoid-s3980" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3981" xml:space="preserve">Nam ſi fiat vt I N, differentia vmbrarum ſiue rectarum,
              <lb/>
            ſiue verſarum, ad A a, differentiam ſtationum: </s>
            <s xml:id="echoid-s3982" xml:space="preserve">Ita B N, maior vmbra recta vel
              <lb/>
            d N, vmbra verſa maior ad aliud, gignetur diſtantia A F, vt demonſtratum eſt
              <lb/>
            Numero 3. </s>
            <s xml:id="echoid-s3983" xml:space="preserve">& </s>
            <s xml:id="echoid-s3984" xml:space="preserve">1.</s>
            <s xml:id="echoid-s3985" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3986" xml:space="preserve">6. </s>
            <s xml:id="echoid-s3987" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            ſi quando in vna ſtatione linea fiduciæ tranſierit per C, aſſumi
              <lb/>
            poterit vel latus vmbrærectæ BC, vel verſæ C D, prout in altera ſtatione abſciſſa
              <lb/>
            erit vmbra recta, vel verſa; </s>
            <s xml:id="echoid-s3988" xml:space="preserve">vt nimirum vmbræ ſint ſimiles.</s>
            <s xml:id="echoid-s3989" xml:space="preserve"/>
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        <div xml:id="echoid-div253" type="section" level="1" n="109">
          <head xml:id="echoid-head112" xml:space="preserve">COROLLARIVM I.</head>
          <p>
            <s xml:id="echoid-s3990" xml:space="preserve">
              <emph style="sc">Colligitvr</emph>
            ex demonſtratis, eundem eſſe operandi modum in vtroq;
              <lb/>
            </s>
            <s xml:id="echoid-s3991" xml:space="preserve">
              <note position="left" xlink:label="note-132-06" xlink:href="note-132-06a" xml:space="preserve">Eundem eſſe
                <lb/>
              modum ope-
                <lb/>
              randi in vtro-
                <lb/>
              quequadrato.</note>
            quadrato: </s>
            <s xml:id="echoid-s3992" xml:space="preserve">quando quidem eædem vmbræ in quadrato pendulo, quæ in ſtabili,
              <lb/>
            abſcinduntur, vt oſtendimus: </s>
            <s xml:id="echoid-s3993" xml:space="preserve">Ita que præcepta, quæ in vno præſcribuntur, in
              <lb/>
            altero quo que obſeruanda ſunt.</s>
            <s xml:id="echoid-s3994" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div255" type="section" level="1" n="110">
          <head xml:id="echoid-head113" xml:space="preserve">COROLLARIVM II.</head>
          <p>
            <s xml:id="echoid-s3995" xml:space="preserve">
              <emph style="sc">Patet</emph>
            etiam ex dictis, operationem non variari, ſiue per vmbras verſas,
              <lb/>
              <note position="left" xlink:label="note-132-07" xlink:href="note-132-07a" xml:space="preserve">Eundem eſſe
                <lb/>
              operandi mo-
                <lb/>
              dum per vm-
                <lb/>
              br{as}
                <unsure/>
              verſ{as}, &
                <lb/>
              per rect{as}.</note>
            ſiue per rectas inſtituatur: </s>
            <s xml:id="echoid-s3996" xml:space="preserve">quando quidem ſemper eſt, vt differentia vmbrarum
              <lb/>
            ad differentiam ſtationum, ita vmbra maior, ad diſtantiam, quæ inueſtiganda
              <lb/>
            proponitur, vt demonſtratum eſt.</s>
            <s xml:id="echoid-s3997" xml:space="preserve"/>
          </p>
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