Cavalieri, Buonaventura, Geometria indivisibilibvs continvorvm : noua quadam ratione promota

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          <p>
            <s xml:id="echoid-s1415" xml:space="preserve">
              <pb o="55" file="0075" n="75" rhead="LIBER I."/>
            AP, KX, quidem ipſis, VG, Λ Υ, &</s>
            <s xml:id="echoid-s1416" xml:space="preserve">, AF, KZ, ipſis, HG,
              <lb/>
            & </s>
            <s xml:id="echoid-s1417" xml:space="preserve">Y, perpendiculares, iunganturque, PE, XT, PF, XZ, &</s>
            <s xml:id="echoid-s1418" xml:space="preserve">, F
              <lb/>
            E, ZT. </s>
            <s xml:id="echoid-s1419" xml:space="preserve">Quoniam ergo, APG, eſt angulus rectus, erit quadra-
              <lb/>
              <note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">47. Primi
                <lb/>
              Elem.</note>
            tum, AG, æquale quadratis, GP, PA, quadratum verò, PA,
              <lb/>
            æquatur duobus quadratis, PE, EA, propter angulum rectum, A
              <lb/>
              <note position="right" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">Defin. 3.
                <lb/>
              Vndec.
                <lb/>
              Elem.</note>
            EP, ergo quadratum, AG, hoc eſt duo quadrata, GE, EA, ęqua-
              <lb/>
            buntur tribus quadratis, GP, PE, EA, & </s>
            <s xml:id="echoid-s1420" xml:space="preserve">ablato communi qua-
              <lb/>
            drato, EA, quadratum, GE, æquabitur quadratis, GP, PE, er-
              <lb/>
            go, EP, erit perpendicularis ipſi, PV, cui etiam eſt perpendicula-
              <lb/>
              <note position="right" xlink:label="note-0075-03" xlink:href="note-0075-03a" xml:space="preserve">48. Primi
                <lb/>
              Elem.</note>
            ris, AP, ergo, APE, erit inclinatio planorum, AV, VH. </s>
            <s xml:id="echoid-s1421" xml:space="preserve">Eo-
              <lb/>
            dem modo oſtendemus, KXT, eſſe inclinationem planorum, Κ Λ,
              <lb/>
              <note position="right" xlink:label="note-0075-04" xlink:href="note-0075-04a" xml:space="preserve">Defin. 6.
                <lb/>
              Vndec.
                <lb/>
              Elem.</note>
            Λ &</s>
            <s xml:id="echoid-s1422" xml:space="preserve">, & </s>
            <s xml:id="echoid-s1423" xml:space="preserve">angu os, EFG, TZY, eſſe rectos. </s>
            <s xml:id="echoid-s1424" xml:space="preserve">Quoniam verò angu-
              <lb/>
            lus, AGV, æquatur ipſi, Κ Υ Λ, (ſunt. </s>
            <s xml:id="echoid-s1425" xml:space="preserve">n. </s>
            <s xml:id="echoid-s1426" xml:space="preserve">figuræ, AV, Κ Λ, ſimi-
              <lb/>
            les ex hypoteſi) etiam, AGP, æquabitur, KYX, &</s>
            <s xml:id="echoid-s1427" xml:space="preserve">, APG KX
              <lb/>
            Y, recti ſunt, ergo triangula, APG, KXY, ſimil a erunt. </s>
            <s xml:id="echoid-s1428" xml:space="preserve">Eodem
              <lb/>
            modo probabimus etiam triangula, AGF, KYZ, eſſe ſimilia, er-
              <lb/>
            go, PG, ad, GA, erit vt, XY, ad, YK, &</s>
            <s xml:id="echoid-s1429" xml:space="preserve">, GA, ad, GF, vt, Y
              <lb/>
            K, ad, YZ, ergo ex æqual@, PG, ad, GF, erit vt, XY, ad, YZ,
              <lb/>
            & </s>
            <s xml:id="echoid-s1430" xml:space="preserve">ſunt latera proportionalia circa æquales augulos, PGF, XYZ,
              <lb/>
            (ſunt.</s>
            <s xml:id="echoid-s1431" xml:space="preserve">n. </s>
            <s xml:id="echoid-s1432" xml:space="preserve">æquales ijs, qui ſunt ad verticem, nempè, HGV, & </s>
            <s xml:id="echoid-s1433" xml:space="preserve">Υ Λ,
              <lb/>
            qui adęquantur, cum ſint ſimilium figurarum, HGV, & </s>
            <s xml:id="echoid-s1434" xml:space="preserve">Υ Λ,) er-
              <lb/>
              <note position="right" xlink:label="note-0075-05" xlink:href="note-0075-05a" xml:space="preserve">6. Sexti
                <lb/>
              Elem.</note>
            go triangula, PGF, XYZ, erunt ſimilia, & </s>
            <s xml:id="echoid-s1435" xml:space="preserve">anguli, GPF, YXZ,
              <lb/>
            vt &</s>
            <s xml:id="echoid-s1436" xml:space="preserve">, GFP, YZX, inter ſe æquales, ergo ipſi, FPE, ZXT; </s>
            <s xml:id="echoid-s1437" xml:space="preserve">P
              <lb/>
            FE, ZXT, inter ſe quoque erunt æquales, cum ſint reſiduirectc-
              <lb/>
            rum, GPE, GFE, YXT, YZT; </s>
            <s xml:id="echoid-s1438" xml:space="preserve">ergo triangula, PEF, XTZ,
              <lb/>
            pariter ſimilia erunt. </s>
            <s xml:id="echoid-s1439" xml:space="preserve">Erit ergo, AP, ad, PG, vt, KX, ad, XY;
              <lb/>
            </s>
            <s xml:id="echoid-s1440" xml:space="preserve">
              <note position="right" xlink:label="note-0075-06" xlink:href="note-0075-06a" xml:space="preserve">4. Sexti
                <lb/>
              Elem.</note>
            PG, ad, PF, vt, XY, ad, XZ; </s>
            <s xml:id="echoid-s1441" xml:space="preserve">&</s>
            <s xml:id="echoid-s1442" xml:space="preserve">, PF, ad, PE, vt, XZ, ad, X
              <lb/>
            T, ergo ex ęquali, AP, ad, PE, erit vt, KX, ad, XT, & </s>
            <s xml:id="echoid-s1443" xml:space="preserve">ſunt an-
              <lb/>
            guli, AEP, KTX, rect@, ergo triangula, APF, KXT, ſitnilia
              <lb/>
              <note position="right" xlink:label="note-0075-07" xlink:href="note-0075-07a" xml:space="preserve">7. Sexti
                <lb/>
              Elem.</note>
            erunt, & </s>
            <s xml:id="echoid-s1444" xml:space="preserve">angu@i, APE, KXT, ęqual@s, qu@@unt inclinationes pla-
              <lb/>
            norum, AV, Κ Λ, ad plana, VH, Λ &</s>
            <s xml:id="echoid-s1445" xml:space="preserve">, ad eandem partein, quod
              <lb/>
            oſſendendum erat.</s>
            <s xml:id="echoid-s1446" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div157" type="section" level="1" n="106">
          <head xml:id="echoid-head117" xml:space="preserve">LEMMA II.</head>
          <p>
            <s xml:id="echoid-s1447" xml:space="preserve">IN eadem antecedentis ſigura ſi @upponamus propoſitas eſſe duas
              <lb/>
            ſimiles quaſcumque rectihneas ſiguras, AV, Κ Λ, interſe, nec-
              <lb/>
              <note position="right" xlink:label="note-0075-08" xlink:href="note-0075-08a" xml:space="preserve">Iux.def.1.
                <lb/>
              S@xti El.</note>
            non, HV, & </s>
            <s xml:id="echoid-s1448" xml:space="preserve">Λ, conuen@entes in homologis lateribus vtriſq; </s>
            <s xml:id="echoid-s1449" xml:space="preserve">com-
              <lb/>
            munibus, GV, Υ Λ, ſint autem homologæ inter ſe, AG, KY; </s>
            <s xml:id="echoid-s1450" xml:space="preserve">H
              <lb/>
            G, & </s>
            <s xml:id="echoid-s1451" xml:space="preserve">Y; </s>
            <s xml:id="echoid-s1452" xml:space="preserve">& </s>
            <s xml:id="echoid-s1453" xml:space="preserve">ipſæ figuræ æquè ad eandem partem inuicem inclinatæ.
              <lb/>
            </s>
            <s xml:id="echoid-s1454" xml:space="preserve">D@co angulos, AGH, KY &</s>
            <s xml:id="echoid-s1455" xml:space="preserve">, ęquales eſſe, & </s>
            <s xml:id="echoid-s1456" xml:space="preserve">circa eo@dem latera
              <lb/>
            pr@portionalia, quod etiam de angulis, DVN, Q Λ ℟, pariter ve-
              <lb/>
            rum eſſe oſtendemus.</s>
            <s xml:id="echoid-s1457" xml:space="preserve"/>
          </p>
        </div>
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