Clavius, Christoph, Geometria practica

Table of contents

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[191.] II. EX diametro aream circuli vera minorem inueſtigare.
[192.] III. EX circumferentia aream circuli vera maiorem colligere.
[193.] IV. EX circumferentia aream circuli vera minorem concludere.
[194.] DE AREA SEGMENTORVM CIRCVLI. Capvt VIII.
[196.] II.
[197.] III.
[198.] IV.
[200.] VI.
[201.] FINIS LIBRI QVARTI.
[202.] GEOMETRIÆ PRACTICÆ LIBER QVINTVS.
[203.] AREAS Solidorum, corporumue perſcrutans.
[204.] DE AREA PARALLELEPIP EDO-rum, Priſmatum, & Cylindrorum. Capvt I.
[205.] DE AREA PYRAMIDVM & Conorum. Capvt II.
[206.] DL AREA FRVSTI PYRA-midis, & Coni. Capvt III.
[207.] SCHOLIVM.
[208.] DE AREA QVINQVE COR-porum regularium. Capvt IV.
[209.] Capvt V.
[210.] PROPOSITIO I.
[211.] COROLLARIVM.
[212.] PROPOSITIO II.
[213.] COROLLARIVM.
[214.] PROPOSITIO III.
[215.] COROLLARIVM.
[216.] PROPOSITIO IV.
[217.] PROPOSITIO V.
[218.] PROPOSITIO VI.
[219.] PROPOSITIO VII.
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          <p>
            <s xml:id="echoid-s13704" xml:space="preserve">
              <pb o="288" file="318" n="318" rhead="GEOMETR. PRACT."/>
            ratorem per radicem illam propinquam partire. </s>
            <s xml:id="echoid-s13705" xml:space="preserve">Vtroque enim modo radix pro-
              <lb/>
              <note position="left" xlink:label="note-318-01" xlink:href="note-318-01a" xml:space="preserve">dratæ & cu-
                <lb/>
              bicæ ex data
                <lb/>
              minutia.</note>
            pinqua fractionis propoſitæ gignetur. </s>
            <s xml:id="echoid-s13706" xml:space="preserve">Et ſi quidem propinqua illa radix nume-
              <lb/>
            ri producti ex numeratore in denominatorem fuerit minor quam vera, reperie-
              <lb/>
            tur priori modo radix fractionis propinqua minor quo que quam vera; </s>
            <s xml:id="echoid-s13707" xml:space="preserve">pro-
              <lb/>
            pterea quod numerus verò minor diuiditur: </s>
            <s xml:id="echoid-s13708" xml:space="preserve">poſteriori verò modo inuenietur
              <lb/>
            radix propinqua fractionis maior quam vera, quod tunc diuiſio fiat per nume-
              <lb/>
            rum vero minorem. </s>
            <s xml:id="echoid-s13709" xml:space="preserve">Contrarium eueniet, ſi radix illa propinqua numeri ex nu-
              <lb/>
            meratore in denominatorem producti fuerit maior quam vera. </s>
            <s xml:id="echoid-s13710" xml:space="preserve">Nam priori mo-
              <lb/>
            do gignetur radix fractionis propinqua maior, quam vera, poſteriori vero mo-
              <lb/>
            do minor, quam vera, vt perſpicuum eſt. </s>
            <s xml:id="echoid-s13711" xml:space="preserve">Hanc regulam propoſui quo que libr.
              <lb/>
            </s>
            <s xml:id="echoid-s13712" xml:space="preserve">4. </s>
            <s xml:id="echoid-s13713" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s13714" xml:space="preserve">2. </s>
            <s xml:id="echoid-s13715" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s13716" xml:space="preserve">5. </s>
            <s xml:id="echoid-s13717" xml:space="preserve">ibique eandem demonſtraui. </s>
            <s xml:id="echoid-s13718" xml:space="preserve">Exemplum huius etiam regulæ
              <lb/>
            ibidem habes.</s>
            <s xml:id="echoid-s13719" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13720" xml:space="preserve">
              <emph style="sc">Pro</emph>
            radice verò cubica: </s>
            <s xml:id="echoid-s13721" xml:space="preserve">duc numeratorem in quadratum denominatoris,
              <lb/>
            & </s>
            <s xml:id="echoid-s13722" xml:space="preserve">producti numeri radicem cubicam propinquam diuide per denominatorem:
              <lb/>
            </s>
            <s xml:id="echoid-s13723" xml:space="preserve">Vel duc denominatorem in quadratum numeratoris, & </s>
            <s xml:id="echoid-s13724" xml:space="preserve">per numeri producti
              <lb/>
            radicem cubicam propinquam partire numeratorem. </s>
            <s xml:id="echoid-s13725" xml:space="preserve">Vtroque enim modo
              <lb/>
            propinqua radix propoſitæ minutiæ proueniet. </s>
            <s xml:id="echoid-s13726" xml:space="preserve">Et priori quidem modo, ſi il-
              <lb/>
            la radix cubica propinqua fuerit minor quam vera, reperietur radix propinqua
              <lb/>
            fractionis minor quo que quam vera, propterea quod diuiſio fit numeri ve-
              <lb/>
            ro minoris per denominatorem fractionis: </s>
            <s xml:id="echoid-s13727" xml:space="preserve">Si autemradix illa propinqua fue-
              <lb/>
            rit maior quam vera, gignetur quoque radix propinqua fractionis maior quam
              <lb/>
            vera, quod tunc numerus vero maior per denominatorem fractionis diuidatur. </s>
            <s xml:id="echoid-s13728" xml:space="preserve">
              <lb/>
            Poſteriorivero modo, ſi radix illa cubica propinquior fuerit minor quam vera,
              <lb/>
            producetur radix propinqua fractionis maior quam vera, quod tunc diuiſio
              <lb/>
            fiat per numerum vero minorem: </s>
            <s xml:id="echoid-s13729" xml:space="preserve">At ſi illa radix cubica propinqua fuerit ma-
              <lb/>
            ior quam vera, erit inuenta radix fractionis propinqua minor quam vera, quan-
              <lb/>
            doquidem tunc diuiditur numerator per numerum vero maiorẽ. </s>
            <s xml:id="echoid-s13730" xml:space="preserve">Exemplum
              <lb/>
            in fractione {8/27}. </s>
            <s xml:id="echoid-s13731" xml:space="preserve">habente verã radicem cubicam {2/3}. </s>
            <s xml:id="echoid-s13732" xml:space="preserve">Ducto numeratore 8. </s>
            <s xml:id="echoid-s13733" xml:space="preserve">in 729. </s>
            <s xml:id="echoid-s13734" xml:space="preserve">
              <lb/>
            quadratum denominatoris 27. </s>
            <s xml:id="echoid-s13735" xml:space="preserve">fit numerus 5832. </s>
            <s xml:id="echoid-s13736" xml:space="preserve">cuius radix cubica eſt 18. </s>
            <s xml:id="echoid-s13737" xml:space="preserve">Hæc
              <lb/>
            diuiſa per denominatorem 27. </s>
            <s xml:id="echoid-s13738" xml:space="preserve">facit {18/27}. </s>
            <s xml:id="echoid-s13739" xml:space="preserve">id eſt, {2/3}. </s>
            <s xml:id="echoid-s13740" xml:space="preserve">pro radice cubica fractionis {8/27}. </s>
            <s xml:id="echoid-s13741" xml:space="preserve">
              <lb/>
            Item ducto denominatore 27. </s>
            <s xml:id="echoid-s13742" xml:space="preserve">in 64. </s>
            <s xml:id="echoid-s13743" xml:space="preserve">quadratum numeratoris 8. </s>
            <s xml:id="echoid-s13744" xml:space="preserve">gignitur nume-
              <lb/>
            rus 1728. </s>
            <s xml:id="echoid-s13745" xml:space="preserve">cuius radix cubica eſt 12. </s>
            <s xml:id="echoid-s13746" xml:space="preserve">per quam ſi diuidatur numerator 8. </s>
            <s xml:id="echoid-s13747" xml:space="preserve">fit Quo-
              <lb/>
            tiens {8/12}. </s>
            <s xml:id="echoid-s13748" xml:space="preserve">hoc eſt, {2/3}. </s>
            <s xml:id="echoid-s13749" xml:space="preserve">vt prius, pro radice cubica fractionis {8/27}. </s>
            <s xml:id="echoid-s13750" xml:space="preserve">propoſitæ. </s>
            <s xml:id="echoid-s13751" xml:space="preserve">Alte-
              <lb/>
            rum exemplum in fractione {5/7}. </s>
            <s xml:id="echoid-s13752" xml:space="preserve">non habente veram radicem cubicam. </s>
            <s xml:id="echoid-s13753" xml:space="preserve">Ducto
              <lb/>
            numeratore 5. </s>
            <s xml:id="echoid-s13754" xml:space="preserve">in 49. </s>
            <s xml:id="echoid-s13755" xml:space="preserve">quadratum denominatoris 7. </s>
            <s xml:id="echoid-s13756" xml:space="preserve">fit numerus 245. </s>
            <s xml:id="echoid-s13757" xml:space="preserve">cuius radix
              <lb/>
            cubica propinqua 6 {25/100}. </s>
            <s xml:id="echoid-s13758" xml:space="preserve">(inuenta per appoſitionum duorum ternariorum ci-
              <lb/>
            frarum) diuiſa per denominatorẽ 7. </s>
            <s xml:id="echoid-s13759" xml:space="preserve">facit Quotientem {625/700}. </s>
            <s xml:id="echoid-s13760" xml:space="preserve">hoc eſt, {25/28}. </s>
            <s xml:id="echoid-s13761" xml:space="preserve">. </s>
            <s xml:id="echoid-s13762" xml:space="preserve">pro ra-
              <lb/>
            dice ſractionis {5/7}. </s>
            <s xml:id="echoid-s13763" xml:space="preserve">Item ducto denominatore 7. </s>
            <s xml:id="echoid-s13764" xml:space="preserve">in 25. </s>
            <s xml:id="echoid-s13765" xml:space="preserve">quadratum numeratoris 5. </s>
            <s xml:id="echoid-s13766" xml:space="preserve">
              <lb/>
            fit numerus 175. </s>
            <s xml:id="echoid-s13767" xml:space="preserve">per cuius radicem cubicam 5 {59/100}. </s>
            <s xml:id="echoid-s13768" xml:space="preserve">propinquam inuentam per ap-
              <lb/>
            poſitionẽ 000000. </s>
            <s xml:id="echoid-s13769" xml:space="preserve">ad 175. </s>
            <s xml:id="echoid-s13770" xml:space="preserve">ſi partiamur numeratorem 5. </s>
            <s xml:id="echoid-s13771" xml:space="preserve">inueniemus Quotien-
              <lb/>
            tem {500/559}. </s>
            <s xml:id="echoid-s13772" xml:space="preserve">pro radice cubica propinqua datæ fractionis {5/7}. </s>
            <s xml:id="echoid-s13773" xml:space="preserve">atque ita de aliis. </s>
            <s xml:id="echoid-s13774" xml:space="preserve">Por-
              <lb/>
            ro hoc modo reperitur radix fractionis propinquior, quam per ſuperiorem re-
              <lb/>
            gulam: </s>
            <s xml:id="echoid-s13775" xml:space="preserve">quia hic ſolum vnus error irrepit propter radicem cubicam propin-
              <lb/>
            quam, quæ vera non eſt, manente tam denominatore in priori modo, quam nu-
              <lb/>
            meratore in poſteriori, in propria ſua quantitate; </s>
            <s xml:id="echoid-s13776" xml:space="preserve">at in ſuperioriregula duo
              <lb/>
            interueniunt errores, propter duas radices cubicas propinquas, quæ veræ
              <lb/>
            non ſunt.</s>
            <s xml:id="echoid-s13777" xml:space="preserve"/>
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