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

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          <p>
            <s xml:id="echoid-s1526" xml:space="preserve">
              <pb o="58" file="0078" n="78" rhead="GEOMETRIÆ"/>
            GO, ſint verò in ambitu triangula, ABC, FIG; </s>
            <s xml:id="echoid-s1527" xml:space="preserve">ABH, FIO, H
              <lb/>
            BC, OIG, ergo tria hęc tribus iam dictis ſimilia erunt, ergo & </s>
            <s xml:id="echoid-s1528" xml:space="preserve">ba-
              <lb/>
            ſes, ACH, FGO, ſimiles erunt, nam cum ſit, AC, ad, CB, vt,
              <lb/>
            FG, ad, GI; </s>
            <s xml:id="echoid-s1529" xml:space="preserve">BC, ad, CH, vt, IG, ad, GO, erit ex ęquali, AC,
              <lb/>
            ad, CH, vt, FG, ad, GO, eadem ratione oſtendemus, CH, ad,
              <lb/>
              <note position="left" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">5. Sexti
                <lb/>
              Elem.</note>
            HA, eſſe vt, GO, ad, OF, ex quo habebitur ex ęquali, CA, ad,
              <lb/>
            AH, eſſe vt, GF, ad, FO, ergo triangula, ACH, FGO, ſimilia
              <lb/>
            erunt. </s>
            <s xml:id="echoid-s1530" xml:space="preserve">Eodem modo probabimus triangula, AHD, FLO, ACD,
              <lb/>
            FGL, eſſe ſimilia, ex quo concludemus ipſas pyramides ſimiles eſſe.
              <lb/>
            </s>
            <s xml:id="echoid-s1531" xml:space="preserve">Quod ſi tria triangula ad, B, I, terminantia omnia non ſint in am-
              <lb/>
            bitu, oſtendemus tamen illa eſſe ſimilia, erunt.</s>
            <s xml:id="echoid-s1532" xml:space="preserve">n. </s>
            <s xml:id="echoid-s1533" xml:space="preserve">vel baſes pyrami-
              <lb/>
            dum, quarum tria triangula verticalia erunt in ambitu, vel ſaltem
              <lb/>
            aliarum pyramidum, quarum triangula ſimilia eſſe probabuntur,
              <lb/>
            quia erunt baſes pyramidum tria triangula verticalia in ambitu ha-
              <lb/>
            bentium, ad hæc.</s>
            <s xml:id="echoid-s1534" xml:space="preserve">n. </s>
            <s xml:id="echoid-s1535" xml:space="preserve">tandem deuenire neceſſe erit: </s>
            <s xml:id="echoid-s1536" xml:space="preserve">Igitur oſtenſum
              <lb/>
            eſt, quod proponebatur.</s>
            <s xml:id="echoid-s1537" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div164" type="section" level="1" n="109">
          <head xml:id="echoid-head120" xml:space="preserve">COROLLARIVM.</head>
          <p style="it">
            <s xml:id="echoid-s1538" xml:space="preserve">_Q_Via verò in pyramidibus triangulatis, BAHC, IFGO, exiſten-
              <lb/>
            tibus ſimilibus illarum triangulis verticalibus, baſes, ACH, F
              <lb/>
            GO, neceſſiriò ſimiles eſſe oſtenſæ ſunt, ideò ex hoc colligimus ſi
              <lb/>
            in duabus pyramidibus triangulatis tria verticalia triangulatribus ver-
              <lb/>
            ticalibus triangulis ſimilia ſint, etiam baſes ſimiles eſſe.</s>
            <s xml:id="echoid-s1539" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div165" type="section" level="1" n="110">
          <head xml:id="echoid-head121" xml:space="preserve">LEMMA V.</head>
          <p>
            <s xml:id="echoid-s1540" xml:space="preserve">SI duo ſimilia triangula fuerint ſubiectis planis æquè ad eandem
              <lb/>
            partem inclinata, ita vt communes cum illis ſectiones ſint ea-
              <lb/>
            rum latera homologa, quæ tanquam baſes aſſumantur; </s>
            <s xml:id="echoid-s1541" xml:space="preserve">ab eorum
              <lb/>
            autem verticibus rectæ lineæ in ſublimi fuerint conſtitutæ, angulos
              <lb/>
            æquales cum eorum lateribus homologis continentes, illæ erunt ſu-
              <lb/>
            biectis planis æqualiter inclinatæ, vel eiſdem ambo parallelæ; </s>
            <s xml:id="echoid-s1542" xml:space="preserve">ſi au-
              <lb/>
            tem fuerint inclinatæ, & </s>
            <s xml:id="echoid-s1543" xml:space="preserve">vſque ad ſubiecta plana producantur, iun-
              <lb/>
            ganturq; </s>
            <s xml:id="echoid-s1544" xml:space="preserve">pucta occurſuum cum extremis baſium dictorum triangu-
              <lb/>
            lorum, pariter hinc conſtitutæ pyramides ſimiles erunt.</s>
            <s xml:id="echoid-s1545" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1546" xml:space="preserve">Sint ſimilia triangula, ABD, HPO, ſubiectis planis ęquè incli-
              <lb/>
            nata, in baſibus, BD, PO, à quorum verticibus, A, H, rectæ li-
              <lb/>
            neæ, AC, HN, in ſublimi conſtitutæ contineant cum homologis
              <lb/>
            eorum lateribus angulos æquales, ſint nempè anguli, CAB, NH
              <lb/>
            P, necnon, CAD, VHO, inter ſe æquales. </s>
            <s xml:id="echoid-s1547" xml:space="preserve">Dico ipſas, AC, H
              <lb/>
            N, ſubiectis planis eſſe ęqualiter inclinatas, vel eiſdem ambo </s>
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