Clavius, Christoph, In Sphaeram Ioannis de Sacro Bosco commentarius

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            <s xml:id="echoid-s4634" xml:space="preserve">
              <pb o="96" file="132" n="133" rhead="Comment. in I. Cap. Sphæræ"/>
            5. </s>
            <s xml:id="echoid-s4635" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4636" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4637" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4638" xml:space="preserve">patet: </s>
            <s xml:id="echoid-s4639" xml:space="preserve">in quadrilateris autem figuris omnia latcra habentibus æqualiæ
              <lb/>
            (quoniam neceſſario ſunt parallelogramma, vt in ſcholio propoſ. </s>
            <s xml:id="echoid-s4640" xml:space="preserve">34. </s>
            <s xml:id="echoid-s4641" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4642" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4643" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4644" xml:space="preserve">@-
              <lb/>
              <note position="left" xlink:label="note-132-01" xlink:href="note-132-01a" xml:space="preserve">34. primi.</note>
            ſtendimus) ſinguli oppoſiti inter ſe ſint æquales: </s>
            <s xml:id="echoid-s4645" xml:space="preserve">Idcirco totam hanc propoſitionem
              <lb/>
            in triangulis, & </s>
            <s xml:id="echoid-s4646" xml:space="preserve">quadrilateris figuris ita demonſtrabimus. </s>
            <s xml:id="echoid-s4647" xml:space="preserve">Sit primum triangulum
              <lb/>
            A B C, inter ſibi Iſoperimetra triangula maximum. </s>
            <s xml:id="echoid-s4648" xml:space="preserve">Dico illud æquilaterum eſſe & </s>
            <s xml:id="echoid-s4649" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-132-01" xlink:href="fig-132-01a" number="34">
                <image file="132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/132-01"/>
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            æquiangulum. </s>
            <s xml:id="echoid-s4650" xml:space="preserve">Si enim non
              <lb/>
            eſt æqui
              <unsure/>
            laterum, ſed latera
              <lb/>
            A B, B C, ſunt inæqualia:
              <lb/>
            </s>
            <s xml:id="echoid-s4651" xml:space="preserve">ſi ſuper baſem A C, conſti-
              <lb/>
            tuatur, per propoſ. </s>
            <s xml:id="echoid-s4652" xml:space="preserve">7. </s>
            <s xml:id="echoid-s4653" xml:space="preserve">hu-
              <lb/>
            ius triangulum Iſoſceles
              <lb/>
            A D C, ita ut latera A D,
              <lb/>
            D C, ſimul æqualia ſint la-
              <lb/>
            teribus A B, B C, ſimul,
              <lb/>
            erunt triangula A B C,
              <lb/>
            A D C, Iſoperimetra, atque adeo per propoſ. </s>
            <s xml:id="echoid-s4654" xml:space="preserve">8. </s>
            <s xml:id="echoid-s4655" xml:space="preserve">huius, A D C, maius quàm A B C, quod
              <lb/>
            eſt contra hypotheſim. </s>
            <s xml:id="echoid-s4656" xml:space="preserve">Non ergo inæqualia ſunt latera A B, A C, ſed æqualia. </s>
            <s xml:id="echoid-s4657" xml:space="preserve">Eademq́
              <unsure/>
            . </s>
            <s xml:id="echoid-s4658" xml:space="preserve">
              <lb/>
            ratio eſt de cæteris. </s>
            <s xml:id="echoid-s4659" xml:space="preserve">A E quilaterum ergo eſt triangulum A B C. </s>
            <s xml:id="echoid-s4660" xml:space="preserve">Igitur, ex coroll. </s>
            <s xml:id="echoid-s4661" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s4662" xml:space="preserve">
              <lb/>
            5. </s>
            <s xml:id="echoid-s4663" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4664" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4665" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4666" xml:space="preserve">& </s>
            <s xml:id="echoid-s4667" xml:space="preserve">æquiangulum eſt. </s>
            <s xml:id="echoid-s4668" xml:space="preserve">quod eſt, propoſitum.</s>
            <s xml:id="echoid-s4669" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s4670" xml:space="preserve">
              <emph style="sc">Deinde</emph>
            ſit quadrilaterum A B C D, inter omnia ſibi Iſoperimetra maximum.
              <lb/>
            </s>
            <s xml:id="echoid-s4671" xml:space="preserve">Dico illud eſſe & </s>
            <s xml:id="echoid-s4672" xml:space="preserve">æquilaterum & </s>
            <s xml:id="echoid-s4673" xml:space="preserve">æquiangulum. </s>
            <s xml:id="echoid-s4674" xml:space="preserve">Si enim non eſt æquilaterum, ſint late-
              <lb/>
            ra A B, B C, ſi fieri poteſt, inæqualia, ducaturq́ue recta A C. </s>
            <s xml:id="echoid-s4675" xml:space="preserve">Si igitur, per propoſ. </s>
            <s xml:id="echoid-s4676" xml:space="preserve">7. </s>
            <s xml:id="echoid-s4677" xml:space="preserve">
              <lb/>
            huius, ſuper A C, conſtituatur triangulum A E C, iſoperimetrum triangulo A B C,
              <lb/>
            erit, per propo
              <unsure/>
            ſ. </s>
            <s xml:id="echoid-s4678" xml:space="preserve">8. </s>
            <s xml:id="echoid-s4679" xml:space="preserve">huius, triangulum A E C, maius triangulo A B C, Addito, ergo con
              <lb/>
            muni triangulo A C D, erit quadrilaterum A E C D, maius quadrilatero A B C D. </s>
            <s xml:id="echoid-s4680" xml:space="preserve">
              <lb/>
            quod eſt contra hypotheſim cum A B C D, maximum ponatur. </s>
            <s xml:id="echoid-s4681" xml:space="preserve">Non ergo inæqualia ſunt
              <lb/>
            latera A B, B C, ſed ę
              <unsure/>
            qualia. </s>
            <s xml:id="echoid-s4682" xml:space="preserve">Eademq́. </s>
            <s xml:id="echoid-s4683" xml:space="preserve">ratio eſt de cæteris. </s>
            <s xml:id="echoid-s4684" xml:space="preserve">AEquilatera ergo eſt fi-
              <lb/>
            gura A B C D.</s>
            <s xml:id="echoid-s4685" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s4686" xml:space="preserve">
              <emph style="sc">Sit</emph>
            iam quadrilatera figura A B C D, omnium iſoperimetrarum maxima, æqui-
              <lb/>
            latera, vt oſtenſum eſt, at non æquiangula, ſed anguli B A D, C D A, inæquales ſint.
              <lb/>
            </s>
            <s xml:id="echoid-s4687" xml:space="preserve">Quoniam igitur figura A B C D, cum ſit æquilatera parallelogrammum eſt vt in
              <lb/>
            ſcholio propoſ. </s>
            <s xml:id="echoid-s4688" xml:space="preserve">24. </s>
            <s xml:id="echoid-s4689" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4690" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4691" xml:space="preserve">Eucl
              <unsure/>
            . </s>
            <s xml:id="echoid-s4692" xml:space="preserve">demonſtrauimus; </s>
            <s xml:id="echoid-s4693" xml:space="preserve">ſi educantur ex A, & </s>
            <s xml:id="echoid-s4694" xml:space="preserve">D, duœ linea
              <lb/>
            perpendiculares A H, D G, occurrentes lateri B C, in H, & </s>
            <s xml:id="echoid-s4695" xml:space="preserve">G, erit quoque
              <emph style="sc">AHGd</emph>
            ,
              <lb/>
            parallelogrammum. </s>
            <s xml:id="echoid-s4696" xml:space="preserve">Quia uero latera A B, D C, maiora ſunt lateribus AH, D G,
              <lb/>
              <note position="left" xlink:label="note-132-02" xlink:href="note-132-02a" xml:space="preserve">19. primi.</note>
            producantur hæc, ut fiant rectæ A E, D F, lateribus A B, D C, æquales, iungaturq́;
              <lb/>
            </s>
            <s xml:id="echoid-s4697" xml:space="preserve">recta F F. </s>
            <s xml:id="echoid-s4698" xml:space="preserve">Quo facto, erit figura A E F D, iſoperimetra parallelogrammo A B C D,
              <lb/>
            cum latera A E, DF, lateribus A B, D C, ęqualia ſint, latus uero A D, commune,
              <lb/>
              <note position="left" xlink:label="note-132-03" xlink:href="note-132-03a" xml:space="preserve">34. primi.</note>
            & </s>
            <s xml:id="echoid-s4699" xml:space="preserve">latus E F, lateri B C, æquale, quòd vtrumque æquale ſit lateri oppoſito A D.
              <lb/>
            </s>
            <s xml:id="echoid-s4700" xml:space="preserve">Cum ergo figura A E F D, maior ſit parallelogrammo A H G D, hoc autem æquale
              <lb/>
            ſit parallelogrammo A B C D; </s>
            <s xml:id="echoid-s4701" xml:space="preserve">erit quoque figura A E F D, maior parallelogrammo
              <lb/>
              <note position="left" xlink:label="note-132-04" xlink:href="note-132-04a" xml:space="preserve">35. primi.</note>
            A B C D. </s>
            <s xml:id="echoid-s4702" xml:space="preserve">Quare cum eidem ſit iſoperimetra, non erit A B C D, figura quadrilateræ
              <lb/>
            inter ſibi Iſoperimetras maximam. </s>
            <s xml:id="echoid-s4703" xml:space="preserve">quod eſt contra hypotheſim. </s>
            <s xml:id="echoid-s4704" xml:space="preserve">Non ergo inæquales
              <lb/>
            ſunt anguli B A D, C D A. </s>
            <s xml:id="echoid-s4705" xml:space="preserve">ſed æquales: </s>
            <s xml:id="echoid-s4706" xml:space="preserve">atque adeo cum A B C D, ſit parallelogram-
              <lb/>
            mum, erunt anguli oppoſiti B, C, angulis D, A, æquales, proptereaq́; </s>
            <s xml:id="echoid-s4707" xml:space="preserve">tota figura æ
              <unsure/>
            -
              <lb/>
              <note position="left" xlink:label="note-132-05" xlink:href="note-132-05a" xml:space="preserve">34. primi.</note>
            quiangula erit. </s>
            <s xml:id="echoid-s4708" xml:space="preserve">quod eſt propoſitum.</s>
            <s xml:id="echoid-s4709" xml:space="preserve"/>
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