Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s6316" xml:space="preserve">
              <pb o="161" file="191" n="191" rhead="LIBER QVARTVS."/>
            ac proinde radix quadratihuius numeri erit dicti trianguli area: </s>
            <s xml:id="echoid-s6317" xml:space="preserve">quod erat de-
              <lb/>
            monſtrandum.</s>
            <s xml:id="echoid-s6318" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6319" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            autem ex quadrato ipſius D E, in quadratum ipſius A H, produca-
              <lb/>
            tur quadratusnumerus areætrianguli ABC, in hunc modum demonſtro. </s>
            <s xml:id="echoid-s6320" xml:space="preserve">Quo-
              <lb/>
            niam vt Num. </s>
            <s xml:id="echoid-s6321" xml:space="preserve">2. </s>
            <s xml:id="echoid-s6322" xml:space="preserve">oſtendemus, ex D E, in ſemiſſem lateris A B, producitur area
              <lb/>
            trianguli ADB; </s>
            <s xml:id="echoid-s6323" xml:space="preserve">Et ex eadem D E, hoc eſt, ex DG, in ſemiſſemlateris BC, effi-
              <lb/>
            citur area trianguli B D C; </s>
            <s xml:id="echoid-s6324" xml:space="preserve">Item ex eadem D E, id eſt, ex D F, in ſemiſſem late-
              <lb/>
            ris A C, gignitur area trianguli A D C: </s>
            <s xml:id="echoid-s6325" xml:space="preserve">Quod autem fit ex D E, in ſemiſſes late-
              <lb/>
            rum AB, BC, AC, æquale eſt ei, quod fit ex DE, in AH, ex illis ſemiſsibu
              <unsure/>
            s
              <note symbol="a" position="right" xlink:label="note-191-01" xlink:href="note-191-01a" xml:space="preserve">1. ſecun
                <unsure/>
              di.</note>
            ſlatam. </s>
            <s xml:id="echoid-s6326" xml:space="preserve">fiet propterea area trianguli A B C, ex DE, in AH, ac propterea (con-
              <lb/>
            tractis hiſcelineis ad numeros) quadratus numerus areæ eiuſdem trianguli pro-
              <lb/>
            creabitur ex quadrato ipſius DE, in quadratumipſius A H. </s>
            <s xml:id="echoid-s6327" xml:space="preserve">Quando enim duo
              <lb/>
            numeri ſe mutuo multiplicantes fecerint aliquem, producent eorum quadrati
              <lb/>
            ſe mutuo multiplicantes quadratum illius producti, quod ita perſpicuum fiet.
              <lb/>
            </s>
            <s xml:id="echoid-s6328" xml:space="preserve">Duo numeri A, & </s>
            <s xml:id="echoid-s6329" xml:space="preserve">B, ſemultiplicantes faciant D; </s>
            <s xml:id="echoid-s6330" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s6331" xml:space="preserve">ambo ſeipſos multiplicantes faciant C, & </s>
            <s xml:id="echoid-s6332" xml:space="preserve">E: </s>
            <s xml:id="echoid-s6333" xml:space="preserve">
              <lb/>
            Denique hi quadrati C, & </s>
            <s xml:id="echoid-s6334" xml:space="preserve">E, ſemultiplicantes
              <lb/>
              <figure xlink:label="fig-191-01" xlink:href="fig-191-01a" number="121">
                <image file="191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/191-01"/>
              </figure>
            faciant F. </s>
            <s xml:id="echoid-s6335" xml:space="preserve">Dico F, eſſe quadratum ipſius D.
              <lb/>
            </s>
            <s xml:id="echoid-s6336" xml:space="preserve">Cum enim A, multiplicans ſeipſum, & </s>
            <s xml:id="echoid-s6337" xml:space="preserve">B, faciat
              <lb/>
            C, & </s>
            <s xml:id="echoid-s6338" xml:space="preserve">D: </s>
            <s xml:id="echoid-s6339" xml:space="preserve"> erit vt A, ad B, ita C, ad D: </s>
            <s xml:id="echoid-s6340" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-191-02" xlink:href="note-191-02a" xml:space="preserve">17. ſept.</note>
            queratione, cum B, multiplicans A, & </s>
            <s xml:id="echoid-s6341" xml:space="preserve">ſeipſum,
              <lb/>
            faciat D, & </s>
            <s xml:id="echoid-s6342" xml:space="preserve">E, erit vt A, ad B, ita D, ad E: </s>
            <s xml:id="echoid-s6343" xml:space="preserve">ideoque C, D, E, continuè propor-
              <lb/>
            tionales erunt. </s>
            <s xml:id="echoid-s6344" xml:space="preserve"> Quare qui fit ex C, in E, numerus videlicet F, æqualis erit
              <note symbol="c" position="right" xlink:label="note-191-03" xlink:href="note-191-03a" xml:space="preserve">20. ſept.</note>
            qui fit ex D, in ſe: </s>
            <s xml:id="echoid-s6345" xml:space="preserve">ac proinde F, quadratus erit ipſius D. </s>
            <s xml:id="echoid-s6346" xml:space="preserve">Quæ cumita ſint, cum
              <lb/>
            ex DE, in AH, producatur area trianguli A B C, vt oſtendimus, fiet ex quadrato
              <lb/>
            ipſius DE, in quadratum ipſius AH, quadratus numerus areæ eiuſdem triangu-
              <lb/>
            li ABC. </s>
            <s xml:id="echoid-s6347" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s6348" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6349" xml:space="preserve">2. </s>
            <s xml:id="echoid-s6350" xml:space="preserve">
              <emph style="sc">Altera</emph>
            via, qua ex datis lateribus area trianguli colligitur,
              <lb/>
              <note position="right" xlink:label="note-191-04" xlink:href="note-191-04a" xml:space="preserve">Area trian-
                <lb/>
              guli quo pacto
                <lb/>
              aliter ex datis
                <lb/>
              laterib{us} colli-
                <lb/>
              gatur.</note>
            hæc eſt.</s>
            <s xml:id="echoid-s6351" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6352" xml:space="preserve">Ex quouis angulo ad lat{us} oppoſitum, etiam protractum, ſiop{us} eſt, perpendicularis
              <lb/>
            ducatur. </s>
            <s xml:id="echoid-s6353" xml:space="preserve">Hæc enim (ſi ei{us} quantit{as} cognita fuerit) multiplicata in ſemiſſem baſis, ſeu
              <lb/>
            dicti lateris, vel ei{us} ſemiſſis in totam baſem producet aream trianguli, Velſimauis, tota
              <lb/>
            perpendicularis ducta in totam baſem, numerum procreabit, cui{us} ſemiſſis aream trian-
              <lb/>
            guli offeret:</s>
            <s xml:id="echoid-s6354" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6355" xml:space="preserve">
              <emph style="sc">Nam</emph>
            vtlib. </s>
            <s xml:id="echoid-s6356" xml:space="preserve">7. </s>
            <s xml:id="echoid-s6357" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s6358" xml:space="preserve">1. </s>
            <s xml:id="echoid-s6359" xml:space="preserve">demonſtrauimus, eſt area trianguli æqualis rectan-
              <lb/>
            gulo comprehenſo ſub perpendiculari, & </s>
            <s xml:id="echoid-s6360" xml:space="preserve">ſemiſſe baſis, vel ſub ſemiſſe perpen-
              <lb/>
            dicularis, actotabaſe; </s>
            <s xml:id="echoid-s6361" xml:space="preserve">Item ſemiſsi rectanguli ſub perpendiculari, ac tota baſe
              <lb/>
            comprehenſi. </s>
            <s xml:id="echoid-s6362" xml:space="preserve">Cum ergo per cap. </s>
            <s xml:id="echoid-s6363" xml:space="preserve">1. </s>
            <s xml:id="echoid-s6364" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s6365" xml:space="preserve">area rectanguli illius producatur
              <lb/>
            ex multiplicatione vnius lateris circa angulum rectum in alterum: </s>
            <s xml:id="echoid-s6366" xml:space="preserve">hoc eſt, ex
              <lb/>
            perpendiculari ( quæ vnilateriæqualis eſt) in ſemiſſem baſis trianguli, vel
              <note symbol="d" position="right" xlink:label="note-191-05" xlink:href="note-191-05a" xml:space="preserve">34. primi.</note>
            ſemiſſe perpendicularis ( quæ ſemiſsi lateris eſt æqualis) in totam baſem: </s>
            <s xml:id="echoid-s6367" xml:space="preserve">
              <note symbol="e" position="right" xlink:label="note-191-06" xlink:href="note-191-06a" xml:space="preserve">34. primi.</note>
            deniquerectangulum trianguli duplum ex perpendiculariin totam baſem trian-
              <lb/>
            guli: </s>
            <s xml:id="echoid-s6368" xml:space="preserve">conſtat propoſitum.</s>
            <s xml:id="echoid-s6369" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6370" xml:space="preserve">
              <emph style="sc">Magnitvdo</emph>
            autem dictę perpendicularis, ſicuti & </s>
            <s xml:id="echoid-s6371" xml:space="preserve">baſis, in metiendis
              <lb/>
            campis inueſtiganda eſt per catenulam ferream, quodhęc nequeintendatur,
              <lb/>
            neque remittatur, aut certè, ſi omnialatera nota ſint, Geometrice hoc modo.
              <lb/>
            </s>
            <s xml:id="echoid-s6372" xml:space="preserve">Sit tr
              <unsure/>
            iangulum ABC, cuius latus AB, ſit 10. </s>
            <s xml:id="echoid-s6373" xml:space="preserve">& </s>
            <s xml:id="echoid-s6374" xml:space="preserve">B C, 21. </s>
            <s xml:id="echoid-s6375" xml:space="preserve">& </s>
            <s xml:id="echoid-s6376" xml:space="preserve">A C, 17. </s>
            <s xml:id="echoid-s6377" xml:space="preserve"/>
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