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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        <div xml:id="echoid-div489" type="section" level="1" n="178">
          <p>
            <s xml:id="echoid-s7883" xml:space="preserve">
              <pb o="185" file="215" n="215" rhead="LIBER QVARTVS."/>
            quitur neceſſariò, ambitum polygoni minorem eſſelatere m n. </s>
            <s xml:id="echoid-s7884" xml:space="preserve">Cum enim tri-
              <lb/>
            angulum rectangulum, cuius altitudo ſemidiametro polygoni, & </s>
            <s xml:id="echoid-s7885" xml:space="preserve">baſis ambi-
              <lb/>
            tui æqualis eſt, æquale ſit, ex ſcholio propoſ. </s>
            <s xml:id="echoid-s7886" xml:space="preserve">41. </s>
            <s xml:id="echoid-s7887" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7888" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7889" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s7890" xml:space="preserve">rectangulo ſub ea-
              <lb/>
            dem ſemidiametro, & </s>
            <s xml:id="echoid-s7891" xml:space="preserve">ſemiſſe ambitus polygoni comprehenſa; </s>
            <s xml:id="echoid-s7892" xml:space="preserve">hoc autem, per
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s7893" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7894" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7895" xml:space="preserve">7. </s>
            <s xml:id="echoid-s7896" xml:space="preserve">huius de Iſo perimetris, polygono æquale: </s>
            <s xml:id="echoid-s7897" xml:space="preserve">erit quoque trian-
              <lb/>
            gulum illud minus triangulo l m n. </s>
            <s xml:id="echoid-s7898" xml:space="preserve">Quare cum hæc triangula habeant æquales
              <lb/>
            altitudines; </s>
            <s xml:id="echoid-s7899" xml:space="preserve"> erit vtillud triangulum ad l m n, ita baſis illius ad baſem m n: </s>
            <s xml:id="echoid-s7900" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-215-01" xlink:href="note-215-01a" xml:space="preserve">1. ſexti.</note>
            proinde illa baſis, hoc eſt, ambitus polygoni, baſe m n, minor erit. </s>
            <s xml:id="echoid-s7901" xml:space="preserve">Non ergo
              <lb/>
            ponere potes baſem trianguli l m n, ſi maior eſt, quam peripheria circuli, mino-
              <lb/>
            rem ambitu polygoni: </s>
            <s xml:id="echoid-s7902" xml:space="preserve">In demonſtratione autem Archimedis conſtat, ambitũ
              <lb/>
            polygonimaiorem eſſe baſe trianguli F G H, ſi G H, æqualis eſt peripheriæ cir-
              <lb/>
            culi, cum maior ſit, quam perip heria: </s>
            <s xml:id="echoid-s7903" xml:space="preserve">ac propterea rectè concluſit Archimedes,
              <lb/>
            polygonum eſſe maius triangulo FGH, cum tamen ex hypotheſi aduerſarij o-
              <lb/>
            ſtenſum ſit eſſe minus. </s>
            <s xml:id="echoid-s7904" xml:space="preserve">Itaque potuiſſet Archimedes ita quo que propoſitum
              <lb/>
            colligere. </s>
            <s xml:id="echoid-s7905" xml:space="preserve">Polygonum minus eſt triangulo F G H, propter relictas ſectiones
              <lb/>
            minores magnitudine z. </s>
            <s xml:id="echoid-s7906" xml:space="preserve">Ergo eius ambitus minor eſt baſe G H, (quemadmo-
              <lb/>
            dum proximè demonſtrauimus.) </s>
            <s xml:id="echoid-s7907" xml:space="preserve">hoc eſt, peripheria circuli. </s>
            <s xml:id="echoid-s7908" xml:space="preserve">quod eſt abſurdũ,
              <lb/>
            cum ambitus polygoni maior ſit, quam peripheria. </s>
            <s xml:id="echoid-s7909" xml:space="preserve">Quod abſurdum, doctiſsi-
              <lb/>
            mè Scaliger, colligere non potes in tuo triangulo l m n, cum ſtatuas baſem mn,
              <lb/>
            perip heria circuli maiorem. </s>
            <s xml:id="echoid-s7910" xml:space="preserve">Et ſane miror te, Mathematicus cũ ſis, negare quã-
              <lb/>
            titatẽ aliquam illi eſſe æqualem, qua neque maior eſt, neque minor. </s>
            <s xml:id="echoid-s7911" xml:space="preserve">Si enim æ-
              <lb/>
            qualis non eſt, erit inæqualis. </s>
            <s xml:id="echoid-s7912" xml:space="preserve">Igitur vel maior vel minor, contra hypotheſim,
              <lb/>
            cum dicatur neque maior eſſe, neque minor. </s>
            <s xml:id="echoid-s7913" xml:space="preserve">An non vides, non ſolum Archi-
              <lb/>
            medem, ſed etiam Euclidem lib. </s>
            <s xml:id="echoid-s7914" xml:space="preserve">12. </s>
            <s xml:id="echoid-s7915" xml:space="preserve">hunc argumentandi modum frequentiſsimè
              <lb/>
            vſurpare?</s>
            <s xml:id="echoid-s7916" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div491" type="section" level="1" n="179">
          <head xml:id="echoid-head186" xml:space="preserve">PROPOSITIO II.</head>
          <p>
            <s xml:id="echoid-s7917" xml:space="preserve">CVIVSLIBET circuli peripheria tripla eſt diametri, & </s>
            <s xml:id="echoid-s7918" xml:space="preserve">adhuc ſupe-
              <lb/>
            rat parte, quæ quidem minor eſt decem ſeptuageſimis, hoc eſt, ſepti-
              <lb/>
            ma parte diametri, maior verò decem ſeptuageſimis primis.</s>
            <s xml:id="echoid-s7919" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7920" xml:space="preserve">
              <emph style="sc">Hæc</emph>
            eſt Archimedis propoſitio 3. </s>
            <s xml:id="echoid-s7921" xml:space="preserve">quam nos ſecundam facimus, vt do ctri-
              <lb/>
            næ ordo ſeruetur, quando quidem ſequens propoſitio 3. </s>
            <s xml:id="echoid-s7922" xml:space="preserve">quamipſe 2. </s>
            <s xml:id="echoid-s7923" xml:space="preserve">facit, hãc
              <lb/>
            noſtram propoſitionem 2. </s>
            <s xml:id="echoid-s7924" xml:space="preserve">in demonſtrationem adhibet.</s>
            <s xml:id="echoid-s7925" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7926" xml:space="preserve">Sit igitur circulus ABCD, cuius centrum E, diameter AB, quam ad rectos an-
              <lb/>
            gulos ſecet ſemidiameter E c, & </s>
            <s xml:id="echoid-s7927" xml:space="preserve">e c F, ad E c, perpendicularis ducatur, quæ
              <note symbol="b" position="right" xlink:label="note-215-02" xlink:href="note-215-02a" xml:space="preserve">16. tertij.</note>
            culum tangetin c. </s>
            <s xml:id="echoid-s7928" xml:space="preserve">Ducatur latus hexagoni AD, quod ſemidiametro æquale
              <note symbol="c" position="right" xlink:label="note-215-03" xlink:href="note-215-03a" xml:space="preserve">15. quar.</note>
            rit, & </s>
            <s xml:id="echoid-s7929" xml:space="preserve">arcus AD, grad. </s>
            <s xml:id="echoid-s7930" xml:space="preserve">60. </s>
            <s xml:id="echoid-s7931" xml:space="preserve">Ideoq; </s>
            <s xml:id="echoid-s7932" xml:space="preserve">D c. </s>
            <s xml:id="echoid-s7933" xml:space="preserve">grad. </s>
            <s xml:id="echoid-s7934" xml:space="preserve">30. </s>
            <s xml:id="echoid-s7935" xml:space="preserve">Ducta ergo recta E D e, erit angu-
              <lb/>
            lus e E c, tertia pars recti, cum rectus angulus contineat grad. </s>
            <s xml:id="echoid-s7936" xml:space="preserve">90. </s>
            <s xml:id="echoid-s7937" xml:space="preserve">Fiat quo que
              <lb/>
            angulus c E F, angulo c E e, æqualis: </s>
            <s xml:id="echoid-s7938" xml:space="preserve">eruntq; </s>
            <s xml:id="echoid-s7939" xml:space="preserve">angulie, F, inter ſe æquales, quod
              <lb/>
            vterque complementum ſit tertiæ partis angulirecti, ac proinde vter que duas
              <lb/>
            tertias partes vnius recti comprehendet. </s>
            <s xml:id="echoid-s7940" xml:space="preserve"> Cum ergo omnes tres anguliin
              <note symbol="d" position="right" xlink:label="note-215-04" xlink:href="note-215-04a" xml:space="preserve">32. primi.</note>
            gulo e E F, contineant {6/3}. </s>
            <s xml:id="echoid-s7941" xml:space="preserve">vnius recti, continebit quo que e E F, {2/3}. </s>
            <s xml:id="echoid-s7942" xml:space="preserve">vnius recti
              <lb/>
            ipſumq; </s>
            <s xml:id="echoid-s7943" xml:space="preserve">triangulum æquiangulum erit, hoc eſt, per coroll. </s>
            <s xml:id="echoid-s7944" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s7945" xml:space="preserve">6. </s>
            <s xml:id="echoid-s7946" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7947" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7948" xml:space="preserve">Euc.
              <lb/>
            </s>
            <s xml:id="echoid-s7949" xml:space="preserve">æquilaterũ; </s>
            <s xml:id="echoid-s7950" xml:space="preserve">proptereaq; </s>
            <s xml:id="echoid-s7951" xml:space="preserve">perpendicularis E c, baſem e F, bifariã ſecabit, ex ſcho-
              <lb/>
            lio propoſ. </s>
            <s xml:id="echoid-s7952" xml:space="preserve">26. </s>
            <s xml:id="echoid-s7953" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7954" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7955" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s7956" xml:space="preserve">atq; </s>
            <s xml:id="echoid-s7957" xml:space="preserve">ideo E@e, ipſius e c, dupla erit. </s>
            <s xml:id="echoid-s7958" xml:space="preserve">Poſita igitur c </s>
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