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

Table of handwritten notes

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              <pb o="167" file="0187" n="187" rhead="LIBER II."/>
            OX, vel vt aliæ figuræ ſimiles ab ipſis deſcriptæ, ſiue abipſis ſimplici-
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            bus, ſiue ab ipſis adiuncta quadam linea, vnde caſus iſte ad caſum Theo-
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            rematis præſentis, vel antecedentis deductus erit, & </s>
            <s xml:id="echoid-s4001" xml:space="preserve">ideò patebit, om-
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            nes lineas, AE, ad omnes lineas trianguli, BCE, vel omnes figuras ſi-
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            miles, AE, ad omnes figuras ſimiles trianguli, BCE, ideſt vel maxi-
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            mas abſciſſarum, BE, ad abſciſſas omnes ipſius, BE, vel earum figuras
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            ſimiles eſſe, vt omnia quadrata, B F, ad omnia quadrata figuræ, BEF.
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            </s>
            <s xml:id="echoid-s4002" xml:space="preserve">Vocabuntur autem iſtæ; </s>
            <s xml:id="echoid-s4003" xml:space="preserve">Quatuor ordinum magnitudines collectæ iuxta
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            quatuor magnitudines proportionales vtcunque inuentas, quæ fuerunt
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            ex. </s>
            <s xml:id="echoid-s4004" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s4005" xml:space="preserve">prima quadratum, OQ, ſecunda quadratum, OP, tertia, EB,
              <lb/>
            quarta, BO, magnitudines autem collecta iuxta primam. </s>
            <s xml:id="echoid-s4006" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s4007" xml:space="preserve">ex. </s>
            <s xml:id="echoid-s4008" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s4009" xml:space="preserve">om
              <lb/>
            nia quadrata, BF, dicentur primi ordinis, collectæ verò iuxta ſecun-
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            dam. </s>
            <s xml:id="echoid-s4010" xml:space="preserve">ſ. </s>
            <s xml:id="echoid-s4011" xml:space="preserve">omnia quadrata figuræ, BEF, magnitudines ſecundi ordinis, col-
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            lectæ verò iuxta tertiam magnitudines tertij ordinis, & </s>
            <s xml:id="echoid-s4012" xml:space="preserve">tandem collectæ
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            iuxta quartam magnitudines quarti ordinis, ſic igitur appellabimus hos
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            quatuor magnitudinum ordines. </s>
            <s xml:id="echoid-s4013" xml:space="preserve">In ſupradictis autem, quod dicimus de
              <lb/>
            abſciſſis, idem intellige de reſiduis abſciſſarum, & </s>
            <s xml:id="echoid-s4014" xml:space="preserve">quod de ipſis ſimpli-
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            cibus, idem de eiſdem adiunctis alijs, ſiue ſint recti, ſiue eiuſdem obli-
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            qui tranſitus: </s>
            <s xml:id="echoid-s4015" xml:space="preserve">Hoc autem Corollarium præ cæteris ſummè animaduerten-
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            dumeſt, ac memoriæ diligentiſſimè commendandum, ex hoc enim potiſ-
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            ſimas demonſtrationes tanquam ex fonte dermari ſtudioſus in ſequen-
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            tium Librorum lectione ſacilè comprehendet.</s>
            <s xml:id="echoid-s4016" xml:space="preserve"/>
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        <div xml:id="echoid-div416" type="section" level="1" n="253">
          <head xml:id="echoid-head268" xml:space="preserve">THEOREMA XXVII. PROPOS. XXVII.</head>
          <p>
            <s xml:id="echoid-s4017" xml:space="preserve">SI duo trapezia fuerint in eadem baſi, ſumpto vnolate-
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            rum &</s>
            <s xml:id="echoid-s4018" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4019" xml:space="preserve">quidiſtantium pro baſi, & </s>
            <s xml:id="echoid-s4020" xml:space="preserve">regula, & </s>
            <s xml:id="echoid-s4021" xml:space="preserve">fuerint etiam
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            in eadem altitudine reſpectu illius baſis, & </s>
            <s xml:id="echoid-s4022" xml:space="preserve">latera baſi paral-
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            lela fuerint &</s>
            <s xml:id="echoid-s4023" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4024" xml:space="preserve">qualia, trapezia erunt &</s>
            <s xml:id="echoid-s4025" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4026" xml:space="preserve">qualia, & </s>
            <s xml:id="echoid-s4027" xml:space="preserve">omnia eo-
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            rundem quadrata erunt &</s>
            <s xml:id="echoid-s4028" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4029" xml:space="preserve">qualia.</s>
            <s xml:id="echoid-s4030" xml:space="preserve"/>
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            <s xml:id="echoid-s4031" xml:space="preserve">Sint duo trapezia, AERB, IABD, in eadem baſi, AB, quæ
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            ſit ſumpta pro regula, cuiq; </s>
            <s xml:id="echoid-s4032" xml:space="preserve">latera, ER, ID, ſint parallela, & </s>
            <s xml:id="echoid-s4033" xml:space="preserve">in-
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            ter ſe &</s>
            <s xml:id="echoid-s4034" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4035" xml:space="preserve">qualia, Dico trapezia eſſe &</s>
            <s xml:id="echoid-s4036" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4037" xml:space="preserve">qualia, & </s>
            <s xml:id="echoid-s4038" xml:space="preserve">omnia eorundem qua-
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            drata eſſe &</s>
            <s xml:id="echoid-s4039" xml:space="preserve">ae;</s>
            <s xml:id="echoid-s4040" xml:space="preserve">qualia. </s>
            <s xml:id="echoid-s4041" xml:space="preserve">Producantur, AE, BR, donec ſibi occurrant,
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            vt in, O, &</s>
            <s xml:id="echoid-s4042" xml:space="preserve">, AI, BD, donec ſimul incidant, vt in, X, & </s>
            <s xml:id="echoid-s4043" xml:space="preserve">iunga-
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            tur, OX, quia ergo, ER, parallela eſt ipſi, AB, erunt triangula,
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              <note position="right" xlink:label="note-0187-01" xlink:href="note-0187-01a" xml:space="preserve">Iux. diff. 1.
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              Sexti Ele-
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              ment.</note>
            AOB, EOR, ſimilia, & </s>
            <s xml:id="echoid-s4044" xml:space="preserve">eadem ratione ſimilia erunt triangula, A
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            XB, IXD, ergo vt, AB, ad, ER, velad, ID, illiæqualem, ita
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            erit, BO, ad, OR, vt autem, AB, ad, ID, ita eſt, BX, ad, XD,
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              <note position="right" xlink:label="note-0187-02" xlink:href="note-0187-02a" xml:space="preserve">4. Sex. El.</note>
            ergo vt, BO, ad, OR, ita eſt, BX, ad, XD, ergo, OX, </s>
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