Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s11541" xml:space="preserve">
              <pb o="249" file="279" n="279" rhead="LIBER SEXTVS."/>
            proportio GQ.</s>
            <s xml:id="echoid-s11542" xml:space="preserve">ad QI@quæ O, ad P: </s>
            <s xml:id="echoid-s11543" xml:space="preserve">quoniam punctum diuiſionis Q. </s>
            <s xml:id="echoid-s11544" xml:space="preserve">eadit in ſe-
              <lb/>
            cundam partem HI, totius lineę GI, diuidemus AC, baſem ſecundi trianguli da-
              <lb/>
            to puncto F; </s>
            <s xml:id="echoid-s11545" xml:space="preserve">oppoſitam in R, vt eadem ſit proportio AR, ad R C, quæ H Q, ad
              <lb/>
            QI. </s>
            <s xml:id="echoid-s11546" xml:space="preserve">Dico ductam rectam ER, problema efficere, hoc eſt, ita eſſe trapezium AB-
              <lb/>
            FR, ad triangulum RFC, vt O, ad P. </s>
            <s xml:id="echoid-s11547" xml:space="preserve">Quoniam enim rectæ GH, HI, repertæ ſunt
              <lb/>
            triangulis ABF, AFC, proportionales; </s>
            <s xml:id="echoid-s11548" xml:space="preserve">& </s>
            <s xml:id="echoid-s11549" xml:space="preserve">tam ſecundam partem HI, in Q. </s>
            <s xml:id="echoid-s11550" xml:space="preserve">quam
              <lb/>
            ſecundum triangulum AFC, per rectam FR, ſecuimus proportionaliter: </s>
            <s xml:id="echoid-s11551" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-279-01" xlink:href="note-279-01a" xml:space="preserve">1. ſexti.</note>
            ſit triangulum AFR, ad triangulum CFR, vt AR, ad RC, hoc eſt, vt HQ. </s>
            <s xml:id="echoid-s11552" xml:space="preserve">ad QI:
              <lb/>
            </s>
            <s xml:id="echoid-s11553" xml:space="preserve"> Erit vt GQ, ad QI. </s>
            <s xml:id="echoid-s11554" xml:space="preserve">hoc eſt, vt O, ad P, ita trapezium ABFR, ad triangulũ RFC.</s>
            <s xml:id="echoid-s11555" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-279-02" xlink:href="note-279-02a" xml:space="preserve">1. hui{us}.</note>
            quod eſt propoſitum.</s>
            <s xml:id="echoid-s11556" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11557" xml:space="preserve">
              <emph style="sc">Iam</emph>
            verò ſi antecedens proportionis vergere debeat verſus C, proportio-
              <lb/>
            que data ſit O, ad P; </s>
            <s xml:id="echoid-s11558" xml:space="preserve">fiet id commodiſsimè, ſi triangulum ex F, diuidatur ſecũ-
              <lb/>
            dum proportionem P, ad O, ita vt antecedens vergat verſus B, ſicut docuimus.
              <lb/>
            </s>
            <s xml:id="echoid-s11559" xml:space="preserve">Nam tunc pars verſus C, ad reliquam habebit proportionem, quam O, ad P, per
              <lb/>
            conuerſam proportionalitatem. </s>
            <s xml:id="echoid-s11560" xml:space="preserve">Quod etiam in alijs figuris intelligi volo.</s>
            <s xml:id="echoid-s11561" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11562" xml:space="preserve">
              <emph style="sc">Sit</emph>
            deinde multilatera figura quæ cunque ABCDEF, per rectam ex angu-
              <lb/>
            lo A, ductam ſecanda in duas partes, ita vt pars ad B, vergens ad reliquam par-
              <lb/>
            tem proportionem habeat datam M, ad N. </s>
            <s xml:id="echoid-s11563" xml:space="preserve">Ductis ex dato angulo A, ad omnes
              <lb/>
              <note symbol="c" position="right" xlink:label="note-279-03" xlink:href="note-279-03a" xml:space="preserve">3. hui{us}.</note>
            angulos oppoſitos rectis partientibus figuram in quatuortriangula; </s>
            <s xml:id="echoid-s11564" xml:space="preserve">inueni- anturipſis quatu or rectæ proportionales GH, HI, IK, KL. </s>
            <s xml:id="echoid-s11565" xml:space="preserve">Tota deinde GL, ſe-
              <lb/>
            ceturin O, vt eadem ſit proportio G O, ad O L, quæ M, ad N. </s>
            <s xml:id="echoid-s11566" xml:space="preserve">Et quoniam di-
              <lb/>
            uiſionis punctum O, cadit in tertiam lineam IK, ſecabimus tertij trianguli baſem
              <lb/>
            DE, dato angulo A, oppoſitam in P, vt ſecta eſt IK, in O, ducemuſquerectam
              <lb/>
              <figure xlink:label="fig-279-01" xlink:href="fig-279-01a" number="182">
                <image file="279-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/279-01"/>
              </figure>
            AP. </s>
            <s xml:id="echoid-s11567" xml:space="preserve">Dico eſſerectilineum ABCDPA, ad rectilineum APEFA, vt GO, ad OL,
              <lb/>
            hoc eſt vt M, ad N. </s>
            <s xml:id="echoid-s11568" xml:space="preserve">Quoniam enim triangula rectilinei dati proportionalia ſunt
              <lb/>
            ordine rectis GH, HI, IK, KL, per conſtructionem, tertiumq; </s>
            <s xml:id="echoid-s11569" xml:space="preserve">triangulum ADE,
              <lb/>
            & </s>
            <s xml:id="echoid-s11570" xml:space="preserve">lineam tertiam IK, diuiſimus proportionaliter per rectam A P, & </s>
            <s xml:id="echoid-s11571" xml:space="preserve">in puncto
              <lb/>
            O; </s>
            <s xml:id="echoid-s11572" xml:space="preserve"> cum ſit vt DP, ad PE, hoc eſt, vt IO, ad OK, ita triangulum ADP, ad trian- gulum A P E; </s>
            <s xml:id="echoid-s11573" xml:space="preserve"> Erunt totæ quoque magnitudines ſectæ proportionaliter,
              <note symbol="d" position="right" xlink:label="note-279-04" xlink:href="note-279-04a" xml:space="preserve">1. ſexti.</note>
            eſt, erit ABCDPA, ad APEFA, vt G O, ad O L, hoc eſt, vt M, ad N. </s>
            <s xml:id="echoid-s11574" xml:space="preserve">quod eſt
              <lb/>
              <note symbol="e" position="right" xlink:label="note-279-05" xlink:href="note-279-05a" xml:space="preserve">1. hui{us}.</note>
            propoſitum.</s>
            <s xml:id="echoid-s11575" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11576" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            ſi punctum O, diuidens rectam GL, in duas partes proportionis da-
              <lb/>
            tæ M, ad N, caderet in aliquod punctorum H, I, K, vt in I, terminum ſecundæ li-
              <lb/>
            neæ HI, diuideret recta AD, terminans ſecundum triangulum A C D, totum re-
              <lb/>
            ctilineum in datam proportionem. </s>
            <s xml:id="echoid-s11577" xml:space="preserve">Nam tunc per primam partem propoſ. </s>
            <s xml:id="echoid-s11578" xml:space="preserve">1.
              <lb/>
            </s>
            <s xml:id="echoid-s11579" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s11580" xml:space="preserve">eſſet ABCDA, ad ADEFA, vt M, ad N, vt peſpicuum eſt.</s>
            <s xml:id="echoid-s11581" xml:space="preserve"/>
          </p>
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