Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s6721" xml:space="preserve">
              <pb o="166" file="196" n="196" rhead="GEOMETR. PRACT."/>
            AB, & </s>
            <s xml:id="echoid-s6722" xml:space="preserve">reliquus numerus 880. </s>
            <s xml:id="echoid-s6723" xml:space="preserve">ducatur in 144. </s>
            <s xml:id="echoid-s6724" xml:space="preserve">quadratũ ſemiſsis baſis, erit pro-
              <lb/>
            du @ti 126720. </s>
            <s xml:id="echoid-s6725" xml:space="preserve">radix quadrata 355 {695/711}. </s>
            <s xml:id="echoid-s6726" xml:space="preserve">(quæ paulo minor eſt vera radice) area
              <lb/>
            trianguli ABC. </s>
            <s xml:id="echoid-s6727" xml:space="preserve">Nam ſi quadratum ſemiſsis baſis DC, auferatur ex quadrato la-
              <lb/>
              <note symbol="a" position="left" xlink:label="note-196-01" xlink:href="note-196-01a" xml:space="preserve">47. primi.</note>
            teris AC, reliquum fit quadratum perpendicularis AD, quod ex ſcholio pro- poſ. </s>
            <s xml:id="echoid-s6728" xml:space="preserve">26. </s>
            <s xml:id="echoid-s6729" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s6730" xml:space="preserve">1. </s>
            <s xml:id="echoid-s6731" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s6732" xml:space="preserve">perpendicularis AD, baſem BC, ſecet bifariam in D. </s>
            <s xml:id="echoid-s6733" xml:space="preserve">Quare
              <lb/>
            vt circa finem Num 2. </s>
            <s xml:id="echoid-s6734" xml:space="preserve">oſtendimus, quadratum perpendicularis AD, ductum in
              <lb/>
            quadratum D C, ſemiſsis baſis producet quadratum areæ trianguli A B C. </s>
            <s xml:id="echoid-s6735" xml:space="preserve">Ea-
              <lb/>
            demq; </s>
            <s xml:id="echoid-s6736" xml:space="preserve">ratio eſt in triãgulo æquilatero, cum hoc habeat etiã duo latera æqualia.
              <lb/>
            </s>
            <s xml:id="echoid-s6737" xml:space="preserve">
              <note position="left" xlink:label="note-196-02" xlink:href="note-196-02a" xml:space="preserve">Area trian-
                <lb/>
              guliæ quilate-
                <lb/>
              ri.</note>
            </s>
          </p>
          <p>
            <s xml:id="echoid-s6738" xml:space="preserve">5. </s>
            <s xml:id="echoid-s6739" xml:space="preserve">
              <emph style="sc">Pro</emph>
            area tamen trianguliæ quilateri hæc etiam regula ab auctoribus tra-
              <lb/>
            ditur, quamuis à nemine (quod ſciam) demonſtrata ſit. </s>
            <s xml:id="echoid-s6740" xml:space="preserve">Quadratum lateris du-
              <lb/>
            catur in 13. </s>
            <s xml:id="echoid-s6741" xml:space="preserve">productuſque numerus per 30. </s>
            <s xml:id="echoid-s6742" xml:space="preserve">diuidatur. </s>
            <s xml:id="echoid-s6743" xml:space="preserve">Quotiens enim erit area
              <lb/>
            trianguli æquilateri. </s>
            <s xml:id="echoid-s6744" xml:space="preserve">Vt ſi vnum latus æquilateri trianguli ſit 10. </s>
            <s xml:id="echoid-s6745" xml:space="preserve">ducatur qua-
              <lb/>
            dratum lateris 10. </s>
            <s xml:id="echoid-s6746" xml:space="preserve">nimirum 100, in 13. </s>
            <s xml:id="echoid-s6747" xml:space="preserve">productuſque numerus 1300. </s>
            <s xml:id="echoid-s6748" xml:space="preserve">per 30. </s>
            <s xml:id="echoid-s6749" xml:space="preserve">diui-
              <lb/>
            datur. </s>
            <s xml:id="echoid-s6750" xml:space="preserve">Quotiens enim 43 {1/3}. </s>
            <s xml:id="echoid-s6751" xml:space="preserve">erit area trianguli. </s>
            <s xml:id="echoid-s6752" xml:space="preserve">Hanc regulam ita demonſtro.
              <lb/>
            </s>
            <s xml:id="echoid-s6753" xml:space="preserve">Area trianguli æquilateri, cuius ſingula latera ſunt 1. </s>
            <s xml:id="echoid-s6754" xml:space="preserve">eſt radix quadrata huius
              <lb/>
            numeri {3/16}. </s>
            <s xml:id="echoid-s6755" xml:space="preserve">(Nam perregulam præcedentem Nume. </s>
            <s xml:id="echoid-s6756" xml:space="preserve">4. </s>
            <s xml:id="echoid-s6757" xml:space="preserve">explicatam, ſi {1/4}. </s>
            <s xml:id="echoid-s6758" xml:space="preserve">qua-
              <lb/>
            dratum ſemiſsis lateris dematur ex 1. </s>
            <s xml:id="echoid-s6759" xml:space="preserve">quadrato lateris, & </s>
            <s xml:id="echoid-s6760" xml:space="preserve">reliquus numerus {3/4}. </s>
            <s xml:id="echoid-s6761" xml:space="preserve">
              <lb/>
            ducatur in idem quadratum {1/4}. </s>
            <s xml:id="echoid-s6762" xml:space="preserve">ſemiſsis lateris, producetur quadratum areæ
              <lb/>
            trianguli {3/16}.) </s>
            <s xml:id="echoid-s6763" xml:space="preserve">nimirum {13/30}. </s>
            <s xml:id="echoid-s6764" xml:space="preserve">proximè. </s>
            <s xml:id="echoid-s6765" xml:space="preserve">Cum ergo quadratum lateris 1. </s>
            <s xml:id="echoid-s6766" xml:space="preserve">ad qua-
              <lb/>
            dratum lateris 10. </s>
            <s xml:id="echoid-s6767" xml:space="preserve">hoc eſt, 1. </s>
            <s xml:id="echoid-s6768" xml:space="preserve">ad 100. </s>
            <s xml:id="echoid-s6769" xml:space="preserve">eandem proportionem habeat, quam area
              <lb/>
            {13/30}. </s>
            <s xml:id="echoid-s6770" xml:space="preserve">trianguli, cuius vnumlatus eſt 1. </s>
            <s xml:id="echoid-s6771" xml:space="preserve">ad aream trianguli, cuius vnum latus eſt 10. </s>
            <s xml:id="echoid-s6772" xml:space="preserve">
              <lb/>
            quod vtraque proportio proportionis lateris 1. </s>
            <s xml:id="echoid-s6773" xml:space="preserve">ad latus 10. </s>
            <s xml:id="echoid-s6774" xml:space="preserve">ſit duplicata: </s>
            <s xml:id="echoid-s6775" xml:space="preserve">ſi
              <note symbol="b" position="left" xlink:label="note-196-03" xlink:href="note-196-03a" xml:space="preserve">20. & 19.
                <lb/>
              ſexti.</note>
            at vt 1. </s>
            <s xml:id="echoid-s6776" xml:space="preserve">(quadratum lateris 1.) </s>
            <s xml:id="echoid-s6777" xml:space="preserve">ad 100. </s>
            <s xml:id="echoid-s6778" xml:space="preserve">(quadratum lateris 10.) </s>
            <s xml:id="echoid-s6779" xml:space="preserve">ita area {13/30}. </s>
            <s xml:id="echoid-s6780" xml:space="preserve">ad a-
              <lb/>
            liud, pro ducetur area trianguli, cuius vnum latus eſt 10. </s>
            <s xml:id="echoid-s6781" xml:space="preserve">Hocautem fit, ducen-
              <lb/>
            do ſecundum numerum 100. </s>
            <s xml:id="echoid-s6782" xml:space="preserve">in tertium {13/30}. </s>
            <s xml:id="echoid-s6783" xml:space="preserve">hoc eſt, (vt conſtat ex regula mul-
              <lb/>
            tiplicationis fra ctorum, ducendo 100. </s>
            <s xml:id="echoid-s6784" xml:space="preserve">in numeratorem 13. </s>
            <s xml:id="echoid-s6785" xml:space="preserve">& </s>
            <s xml:id="echoid-s6786" xml:space="preserve">productum per
              <lb/>
            denominatorem 30. </s>
            <s xml:id="echoid-s6787" xml:space="preserve">diuidendo. </s>
            <s xml:id="echoid-s6788" xml:space="preserve">Neque vero opus eſt pro ductum hunc nume-
              <lb/>
            rum 43 {1/3}. </s>
            <s xml:id="echoid-s6789" xml:space="preserve">per primum 1. </s>
            <s xml:id="echoid-s6790" xml:space="preserve">partiri, cum vnitas diuidens, aut multiplicans quem-
              <lb/>
            cunq; </s>
            <s xml:id="echoid-s6791" xml:space="preserve">numerũ producat numerum eundem. </s>
            <s xml:id="echoid-s6792" xml:space="preserve">Sic etiam ſi latus vnum trianguli
              <lb/>
            æquilateri ſit 6. </s>
            <s xml:id="echoid-s6793" xml:space="preserve">ducemus eius quadratum 36. </s>
            <s xml:id="echoid-s6794" xml:space="preserve">in {13/30}. </s>
            <s xml:id="echoid-s6795" xml:space="preserve">hoc eſt in 13. </s>
            <s xml:id="echoid-s6796" xml:space="preserve">numeratorem,
              <lb/>
            productumq; </s>
            <s xml:id="echoid-s6797" xml:space="preserve">468. </s>
            <s xml:id="echoid-s6798" xml:space="preserve">per 30. </s>
            <s xml:id="echoid-s6799" xml:space="preserve">partiemur. </s>
            <s xml:id="echoid-s6800" xml:space="preserve">Quotiens namq; </s>
            <s xml:id="echoid-s6801" xml:space="preserve">15 {3/5}. </s>
            <s xml:id="echoid-s6802" xml:space="preserve">erit trianguli pro-
              <lb/>
            poſiti area.</s>
            <s xml:id="echoid-s6803" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6804" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            autem {13/30}. </s>
            <s xml:id="echoid-s6805" xml:space="preserve">ſitradix quadrata numeri {3/16}, patet ex regula qua cuiuſ-
              <lb/>
              <note position="left" xlink:label="note-196-04" xlink:href="note-196-04a" xml:space="preserve">Radix qua-
                <lb/>
              drata numeri
                <lb/>
              fracti quo pa-
                <lb/>
              cto eruatur.</note>
            uis fracti numeri radix extrahitur: </s>
            <s xml:id="echoid-s6806" xml:space="preserve">quæ talis eſt. </s>
            <s xml:id="echoid-s6807" xml:space="preserve">Numerator in denominatorem
              <lb/>
            ducatur, & </s>
            <s xml:id="echoid-s6808" xml:space="preserve">productiradix propinqua inueniatur. </s>
            <s xml:id="echoid-s6809" xml:space="preserve">Sienim per hanc radicem
              <lb/>
            diuidemus numeratorem: </s>
            <s xml:id="echoid-s6810" xml:space="preserve">velipſam radicem per denominatorem partiemur;
              <lb/>
            </s>
            <s xml:id="echoid-s6811" xml:space="preserve">exibit radix fractionis propoſitæ: </s>
            <s xml:id="echoid-s6812" xml:space="preserve">priori quidem modo maior quam vera, po-
              <lb/>
            ſterioriautem minor, ſi radix illa propinqua producti ex numeratore in deno-
              <lb/>
            minatorem fuerit minor, quam vera: </s>
            <s xml:id="echoid-s6813" xml:space="preserve">quia in priori illo modo fit diuiſio per nu-
              <lb/>
            merum vero minorem, in poſteriori autẽ numerus vero minor diuiditur. </s>
            <s xml:id="echoid-s6814" xml:space="preserve">Quod
              <lb/>
            ſi radixilla propinqua foret maior, quam vera, produceretur priori modo radix
              <lb/>
            fractionis minor, quam vera, poſterioriautem maior, vtliquet. </s>
            <s xml:id="echoid-s6815" xml:space="preserve">Verbi gratia. </s>
            <s xml:id="echoid-s6816" xml:space="preserve">In-
              <lb/>
            uenienda ſitradix quadrata fra ctionis {3/16}. </s>
            <s xml:id="echoid-s6817" xml:space="preserve">quam diximus eſſe quadratum areæ
              <lb/>
            trianguli æquilateri, cuius vnum latus eſt. </s>
            <s xml:id="echoid-s6818" xml:space="preserve">1. </s>
            <s xml:id="echoid-s6819" xml:space="preserve">Ex 3. </s>
            <s xml:id="echoid-s6820" xml:space="preserve">in 16. </s>
            <s xml:id="echoid-s6821" xml:space="preserve">fit numerus 48. </s>
            <s xml:id="echoid-s6822" xml:space="preserve">cu-
              <lb/>
            ius radix propinqua 6 {12/13}. </s>
            <s xml:id="echoid-s6823" xml:space="preserve">minor quam vera, per quam ſi diuidatur numerator 3. </s>
            <s xml:id="echoid-s6824" xml:space="preserve">
              <lb/>
            prodibit radix {13/30}. </s>
            <s xml:id="echoid-s6825" xml:space="preserve">fractionis {3/16}. </s>
            <s xml:id="echoid-s6826" xml:space="preserve">maior, quam vera: </s>
            <s xml:id="echoid-s6827" xml:space="preserve">at ſi radicem eandem pro-
              <lb/>
            pinquam 6 {12/13}. </s>
            <s xml:id="echoid-s6828" xml:space="preserve">partiamur per denominatorem 16. </s>
            <s xml:id="echoid-s6829" xml:space="preserve">reperietur radix {45/104}. </s>
            <s xml:id="echoid-s6830" xml:space="preserve"/>
          </p>
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