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

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            <s xml:id="echoid-s12455" xml:space="preserve">
              <pb o="487" file="0507" n="507" rhead="LIBER VII."/>
            BZ&</s>
            <s xml:id="echoid-s12456" xml:space="preserve">, quod non eſſet ſuperpoſitum, vt ſupra oſtendimus neceſſe
              <lb/>
            eſſe, ponitur autem totam, BZ&</s>
            <s xml:id="echoid-s12457" xml:space="preserve">, eſſe ſuperpoſitam ipſi, CβΛ, er-
              <lb/>
            go ita ſunt ad inuicem ſuperpoſitę, vt neutrius reſidua habeantur,
              <lb/>
            ergo ita ſunt ſuperpoſitę, vt ſibi congruant, ergo figuræ, BZ&</s>
            <s xml:id="echoid-s12458" xml:space="preserve">, C
              <lb/>
            βΛ, inter ſe ſunt æquales.</s>
            <s xml:id="echoid-s12459" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12460" xml:space="preserve">Sint nunc in eodem ſchemate duæ figuræ ſolidæ quæeunque,
              <lb/>
            BZ&</s>
            <s xml:id="echoid-s12461" xml:space="preserve">, CβΛ, in eiſdem planis parallelis, AD, Y4, conſſitutæ, ductis
              <lb/>
            autem quibuſcunq; </s>
            <s xml:id="echoid-s12462" xml:space="preserve">planis, E6, LΣ, præfatis æquidiſtantibus, ſint
              <lb/>
            conceptæ in ſolidis figuræ, quæ iacent in eodem plano, ſemper in-
              <lb/>
            ter ſe æquales, vt, FG, æqualis, HI, &</s>
            <s xml:id="echoid-s12463" xml:space="preserve">, MN, OP, ſimul ſumptæ
              <lb/>
            (ſit enim ſolida figura ex. </s>
            <s xml:id="echoid-s12464" xml:space="preserve">g. </s>
            <s xml:id="echoid-s12465" xml:space="preserve">BZ&</s>
            <s xml:id="echoid-s12466" xml:space="preserve">, intus vtcunq; </s>
            <s xml:id="echoid-s12467" xml:space="preserve">caua ſecundum
              <lb/>
            ſuperficiem, {12/ }, N, {13/ }, O,) æquales ipſi, SV. </s>
            <s xml:id="echoid-s12468" xml:space="preserve">Dico eaſdem ſolidas
              <lb/>
            figuras æquales eſſe. </s>
            <s xml:id="echoid-s12469" xml:space="preserve">Si enim ſolidum, BZ&</s>
            <s xml:id="echoid-s12470" xml:space="preserve">, cum portionibus,
              <lb/>
            ABY, & </s>
            <s xml:id="echoid-s12471" xml:space="preserve">planorũ, AD, Y4, ipſi conterminantibus, ſolido, CβΛ, ita
              <lb/>
            ſuperpoſuerimus, vt planum, AB, ſit in plano, AD, &</s>
            <s xml:id="echoid-s12472" xml:space="preserve">, Y&</s>
            <s xml:id="echoid-s12473" xml:space="preserve">, in pla-
              <lb/>
            no, Y4, oſtendemus (vt fecimus ſuperius circa parallelarum ipſi, A
              <lb/>
            D, conceptas in figuris planis, BZ&</s>
            <s xml:id="echoid-s12474" xml:space="preserve">, CβΛ, portiones) figuras in
              <lb/>
            ſolidis, BZ&</s>
            <s xml:id="echoid-s12475" xml:space="preserve">, CβΛ, conceptas, quæ erant in eodem plano, etiam
              <lb/>
            poſt ſuperpoſitionem manere in eode plano, & </s>
            <s xml:id="echoid-s12476" xml:space="preserve">ideò adhuc ęqua-
              <lb/>
            les eſſe figuras in ſuperpoſitis ſolidis conceptas, & </s>
            <s xml:id="echoid-s12477" xml:space="preserve">ipſis, AD, Y4,
              <lb/>
            parallelas. </s>
            <s xml:id="echoid-s12478" xml:space="preserve">Niſi ergo totum ſolidum toti congruat in prima ſu-
              <lb/>
            perpoſitione, relinquentur reſidua ſolida, vel ex reſiduis compoſi-
              <lb/>
            ta in vtroq; </s>
            <s xml:id="echoid-s12479" xml:space="preserve">ſolido, quæ non erunt ad inuicem ſuperpoſita, cum
              <lb/>
            enim ex. </s>
            <s xml:id="echoid-s12480" xml:space="preserve">g. </s>
            <s xml:id="echoid-s12481" xml:space="preserve">figuræ, QR, ST, æquentur figuræ, SV, dempta com-
              <lb/>
            muni figura, ST, reliqua, QR, æquabitur reliquæ, TV, hocq; </s>
            <s xml:id="echoid-s12482" xml:space="preserve">cõ-
              <lb/>
            tinget in quouis alio plano ipſi, AD, parallelo occurrenteſolidis,
              <lb/>
            C℟Γ, CβΛ, ergo ſemper habentes reſiduum vnius ſolidi, habebimus
              <lb/>
            etiam reſiduum alterius, & </s>
            <s xml:id="echoid-s12483" xml:space="preserve">patebit, iuxta methodum adhibitam in
              <lb/>
            priori parte huius Propoſitionis circa figuras planas, reſidua ſoli-
              <lb/>
            da, vel reſiduorum aggregata ſemper eſſe in eiſdem parallelis pla-
              <lb/>
            nis, vt reſidua, H℟597, ΙΓΛ, 785, eſſe in planis parallelis, E6, Y4,
              <lb/>
            ac æqualiter analoga: </s>
            <s xml:id="echoid-s12484" xml:space="preserve">ſi ergo hæc reſidua adhuc ſuperponantur,
              <lb/>
            ita vt planum, EH, locetur in plano, H6, &</s>
            <s xml:id="echoid-s12485" xml:space="preserve">, Υβ, in, β4, & </s>
            <s xml:id="echoid-s12486" xml:space="preserve">hoc
              <lb/>
            ſempei fieri intelligatur, donec quod ſuperponitur, vt, BZ&</s>
            <s xml:id="echoid-s12487" xml:space="preserve">, totũ
              <lb/>
            fuerit aſſumptum, tandem ipſum totum, BZ&</s>
            <s xml:id="echoid-s12488" xml:space="preserve">, congruet toti, Cβ
              <lb/>
            Λ, niſi enim toto ſolido, BZ&</s>
            <s xml:id="echoid-s12489" xml:space="preserve">, ipſi, CβΛ, ſuperpoſito, @pſa ſibi cõ-
              <lb/>
            gruerent, eſſet aliquod reſiduum vnius, vt ſolidi, CβΛ, ergo etiam
              <lb/>
            eſſet aliquod reſiduum ſolidi, C℟Γ, ſeu, BZ&</s>
            <s xml:id="echoid-s12490" xml:space="preserve">, illudq; </s>
            <s xml:id="echoid-s12491" xml:space="preserve">non eſſet ſu-
              <lb/>
            perpoſitum, quod eſt abſurdum, ponitur enim iam totum ſolidũ,
              <lb/>
            BZ&</s>
            <s xml:id="echoid-s12492" xml:space="preserve">, eſſe ipſi, CβΛ, ſuperpoſitum, non ergo erit aliquod reſiduũ
              <lb/>
            in ipſisſolidis, ergo ſibi congruent, ergo dictæ figuræ ſolidæ, BZ&</s>
            <s xml:id="echoid-s12493" xml:space="preserve">,
              <lb/>
            CβΛ, inter ſe æquales erunt, quæ fuerunt demonſtranda. </s>
            <s xml:id="echoid-s12494" xml:space="preserve"/>
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