Clavius, Christoph, Geometria practica

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            <s xml:id="echoid-s6258" xml:space="preserve">
              <pb o="160" file="190" n="190" rhead="GGOMETR. PRACT."/>
            K M, baſi K C, oſtenſa æqualis; </s>
            <s xml:id="echoid-s6259" xml:space="preserve"> erunt quo que anguli K B M, KBC, æquales.</s>
            <s xml:id="echoid-s6260" xml:space="preserve">
              <note symbol="a" position="left" xlink:label="note-190-01" xlink:href="note-190-01a" xml:space="preserve">8. primi.</note>
            Itaque quoniam duo latera B H, B K, trianguli B H K, duo bus lateribus B L, B K,
              <lb/>
            trianguli B L K, æqualia ſunt, æqualeſq; </s>
            <s xml:id="echoid-s6261" xml:space="preserve">continent angulos ad B, vt oſtendimus,
              <lb/>
            erunt & </s>
            <s xml:id="echoid-s6262" xml:space="preserve">baſes HK, KL, & </s>
            <s xml:id="echoid-s6263" xml:space="preserve">anguli H, L, æquales. </s>
            <s xml:id="echoid-s6264" xml:space="preserve">Cum ergo H, ex
              <note symbol="b" position="left" xlink:label="note-190-02" xlink:href="note-190-02a" xml:space="preserve">4. primi.</note>
            ne rectus ſit, erit quo que L, rectus. </s>
            <s xml:id="echoid-s6265" xml:space="preserve">Quare cum latera KH, KB, trianguli KBH,
              <lb/>
            lateribus KL, KB, trianguli KBL, æqualia ſint, & </s>
            <s xml:id="echoid-s6266" xml:space="preserve">baſis BH, baſi BL, erunt
              <note symbol="c" position="left" xlink:label="note-190-03" xlink:href="note-190-03a" xml:space="preserve">8. primi.</note>
            anguli BKH, BKL, æquales.</s>
            <s xml:id="echoid-s6267" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6268" xml:space="preserve">
              <emph style="sc">Qvoniam</emph>
            autem ex iis, quæ ad prop. </s>
            <s xml:id="echoid-s6269" xml:space="preserve">32. </s>
            <s xml:id="echoid-s6270" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s6271" xml:space="preserve">1. </s>
            <s xml:id="echoid-s6272" xml:space="preserve">Euclidis demonſtrauimus,
              <lb/>
            quatuor anguli quadrilateri BHKL, quatuor rectis ſunt æquales: </s>
            <s xml:id="echoid-s6273" xml:space="preserve">erunt
              <lb/>
            demptis duobus rectis H, L, duo HBL, HKL, duobus rectis æquales; </s>
            <s xml:id="echoid-s6274" xml:space="preserve">ideo que
              <lb/>
            duobus angulis HBL, EBL, æquales, quod hi quo que duobus ſint rectis
              <note symbol="d" position="left" xlink:label="note-190-04" xlink:href="note-190-04a" xml:space="preserve">13. primi.</note>
            quales, ablato que communi HBL, reliquus HKL, reliquo EBL, æqualis erit:
              <lb/>
            </s>
            <s xml:id="echoid-s6275" xml:space="preserve">ac propterea & </s>
            <s xml:id="echoid-s6276" xml:space="preserve">HKB, ipſi E B D, dimidius dimidio, æqualis erit. </s>
            <s xml:id="echoid-s6277" xml:space="preserve">Cum ergo & </s>
            <s xml:id="echoid-s6278" xml:space="preserve">
              <lb/>
            rectus H, recto E, ſit ęqualis, erit quo que reliquus H B K, in triangulo H B
              <note symbol="e" position="left" xlink:label="note-190-05" xlink:href="note-190-05a" xml:space="preserve">32. primi.</note>
            reliquo EDB, in triangulo EDB, æqualis; </s>
            <s xml:id="echoid-s6279" xml:space="preserve">ac proinde triangula BHK, DEB, æ-
              <lb/>
            quiangula erunt. </s>
            <s xml:id="echoid-s6280" xml:space="preserve"> Quapropter erit vt DE, ad EB, ita BH, ad HK; </s>
            <s xml:id="echoid-s6281" xml:space="preserve">at que
              <note symbol="f" position="left" xlink:label="note-190-06" xlink:href="note-190-06a" xml:space="preserve">4. ſexti.</note>
            co ſi lineæ hę ad numeros contrahantur, erit numerus, qui fit ex D E, in H
              <note symbol="g" position="left" xlink:label="note-190-07" xlink:href="note-190-07a" xml:space="preserve">19. ſeptim.</note>
            æqualis numero, qui fit ex EB, in BH. </s>
            <s xml:id="echoid-s6282" xml:space="preserve"> Eandem ergo proportionem
              <note symbol="h" position="left" xlink:label="note-190-08" xlink:href="note-190-08a" xml:space="preserve">7. quinti.</note>
            quadratus ex DE, ad productum ex DE, in HK, & </s>
            <s xml:id="echoid-s6283" xml:space="preserve">ad productum ex EB, in BH.
              <lb/>
            </s>
            <s xml:id="echoid-s6284" xml:space="preserve"> Sedita eſt quadratus ex D E, ad productum ex DE, in HK, vt DE, ad HK:</s>
            <s xml:id="echoid-s6285" xml:space="preserve">
              <note symbol="i" position="left" xlink:label="note-190-09" xlink:href="note-190-09a" xml:space="preserve">17. ſeptim.</note>
            proptera quod DE, multiplicans DE, & </s>
            <s xml:id="echoid-s6286" xml:space="preserve">HK, fecit & </s>
            <s xml:id="echoid-s6287" xml:space="preserve">quadratum ex DE, & </s>
            <s xml:id="echoid-s6288" xml:space="preserve">pro-
              <lb/>
            ductum ex D E, in H K. </s>
            <s xml:id="echoid-s6289" xml:space="preserve">Eritigitur quadratus quo que ex D E, ad productum ex
              <lb/>
            EB, in BH, vt DE, ad HK. </s>
            <s xml:id="echoid-s6290" xml:space="preserve">Vtautem DE, ad HK, ita eſt AE, ad AH. </s>
            <s xml:id="echoid-s6291" xml:space="preserve"> Nam
              <note symbol="k" position="left" xlink:label="note-190-10" xlink:href="note-190-10a" xml:space="preserve">28. primi.</note>
            parallelæ ſint DE, HK, æquiangula erunt triangula AED, AHK, ex coroll. </s>
            <s xml:id="echoid-s6292" xml:space="preserve">prop.
              <lb/>
            </s>
            <s xml:id="echoid-s6293" xml:space="preserve">4. </s>
            <s xml:id="echoid-s6294" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s6295" xml:space="preserve">6. </s>
            <s xml:id="echoid-s6296" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s6297" xml:space="preserve"> Igitur erit vt AE, ad ED, ita AH, ad HK, & </s>
            <s xml:id="echoid-s6298" xml:space="preserve">permutando,
              <note symbol="l" position="left" xlink:label="note-190-11" xlink:href="note-190-11a" xml:space="preserve">4. ſexti.</note>
            AE, ad AH, ita ED, ad HK. </s>
            <s xml:id="echoid-s6299" xml:space="preserve">Igitur erit quadratus quo que ex D E, ad produ-
              <lb/>
            ctum ex EB, in BH, vt AE, ad AH. </s>
            <s xml:id="echoid-s6300" xml:space="preserve"> Qui ergo fit ex quadrato ipſius DE,
              <note symbol="m" position="left" xlink:label="note-190-12" xlink:href="note-190-12a" xml:space="preserve">19. ſept.</note>
            AH, æqualis erit ei, qui fit AE, in productum ex EB, in BH. </s>
            <s xml:id="echoid-s6301" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s6302" xml:space="preserve">numerus,
              <lb/>
            qui ex producto ex quadrato ipſius D E, in A H, multiplicato in A H, gignitur,
              <lb/>
            æqualis erit numero, qui ex producto ex AE, in productum ex EB, in BH, mul-
              <lb/>
            tiplicato in eundem A H, procreatur. </s>
            <s xml:id="echoid-s6303" xml:space="preserve">(Nam quia ęquales numeri, nimirum
              <lb/>
            productus ex quadrato ipſius DE, in AH, & </s>
            <s xml:id="echoid-s6304" xml:space="preserve">productus ex AE, in productum ex
              <lb/>
            EB, in BH, eundem numerum AH, multiplicant habebunt producti,
              <note symbol="n" position="left" xlink:label="note-190-13" xlink:href="note-190-13a" xml:space="preserve">18. ſept.</note>
            numerus, qui ex producto ex quadrato ipſius DE, in AH, multiplicato in AH,
              <lb/>
            gignitur, & </s>
            <s xml:id="echoid-s6305" xml:space="preserve">numerus, qui ex producto ex AE, in productum ex EB, in BH, mul-
              <lb/>
            tiplicato in eundem A H, procreatur, eandem proportionem, quam multipli-
              <lb/>
            cantes. </s>
            <s xml:id="echoid-s6306" xml:space="preserve">Cum ergo hiæquales ſint, erunt & </s>
            <s xml:id="echoid-s6307" xml:space="preserve">illi producti æquales) hoc eſt, nu-
              <lb/>
            merus productus ex AH, in AH, id eſt, quadratus ipſius AH, ductus in quadra-
              <lb/>
            tum ipſius DE, (Per ſcholium enim propoſ. </s>
            <s xml:id="echoid-s6308" xml:space="preserve">19. </s>
            <s xml:id="echoid-s6309" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s6310" xml:space="preserve">8. </s>
            <s xml:id="echoid-s6311" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s6312" xml:space="preserve">quomodocun-
              <lb/>
            que tres numeri inter ſe multiplicentur, idem ſemper numerus procreatur)
              <lb/>
            æqualis erit numero, qui ex producto ex A E, in productum ex E B, in B H,
              <lb/>
            nempe ex producto trium differentiarum A E, E B, B H, inter ſe multipli-
              <lb/>
            catarum, ducto in A H, id eſt, in ſemiſſem laterum gignitur. </s>
            <s xml:id="echoid-s6313" xml:space="preserve">At ex quadra-
              <lb/>
            to ipſius D E, in quadratum ipſius A H, producitur quadratus numerus areæ
              <lb/>
            trianguli A B C, vt mox oſtendemus. </s>
            <s xml:id="echoid-s6314" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s6315" xml:space="preserve">ex producto trium exceſ-
              <lb/>
            ſuum A E, EB, B H, inter ſe multiplicatorum, ducto in A H, ſemiſſem late-
              <lb/>
            rum A B, B C, A C, producitur idem quadratus numerus areæ @@ianguli A B C:</s>
            <s xml:id="echoid-s6316" xml:space="preserve"/>
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