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

Table of handwritten notes

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            <s xml:id="echoid-s11471" xml:space="preserve">
              <pb o="445" file="0465" n="465" rhead="LIBER VI."/>
            M4, P6, ℟k, pariter compoſita, ita vt circumſcripta ſuperet inſcri-
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            ptam minori ſpatio, quam ſit differentia dictarum figurarum (quę
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            differentia ſit ſpatium, Ω,) igitur trilineum, H℟FK, minori quã-
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            titate ſuperabit figuram inſcriptam, quam ſpatium reſiduum por-
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            tionis circuli, VHC, ergo figura inſcripta erit maior dicto reſiduo,
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            quod eſt abſurdum, nam ſi, AC, diuidamus ſimiliter, vt, KH,
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            in punctis, IBD, & </s>
            <s xml:id="echoid-s11472" xml:space="preserve">deſcripſerimus per eadem puncta ſuper centro,
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            A, circumferentias, INS, BRZ, DΠΟΣ, oſtendemus, vt in ante-
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            cedenti figuram compoſitam ex faſcijs, ΙΒβ, ΒDΔ, DCΧΦ, eſſe
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            æqualem figuræ inſcriptæ trilineo, H℟Fk, & </s>
            <s xml:id="echoid-s11473" xml:space="preserve">conſequenter eſſe
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            maiorem ſpatio reſiduo portionis circuli, VHC, cui tamen inſcri-
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            bitur, quodeſt abſurdum.</s>
            <s xml:id="echoid-s11474" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11475" xml:space="preserve">Sit nunc trilineum, H℟Fk, minus eodem, Ω, dicto reſiduo, & </s>
            <s xml:id="echoid-s11476" xml:space="preserve">
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            cætera, vt prius conſtructa, quia ergo circumſcripta figura ſuperat
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            inſcriptã minorr quantitate, quam ſit, Ω, ſuperabit ipſum trilineũ,
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            H℟FK, multò minori quantitate, ergo figura circumſcripta mi-
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            nor erit ſpatio reſiduo portionis circuli, VHC, oſtendemus autem,
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            vt ſupra figuram compoſitam ex ſectore, ANI, & </s>
            <s xml:id="echoid-s11477" xml:space="preserve">ex faſcijs, IBR,
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            BDO, DCV, eſſe æqualem figuræ circumſcriptæ trilineo, H℟Fk,
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            ergo erit minor ſpatio reſiduò iam dicto, cui tamen circunſcribitur
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            quod eſt abſurdum, trilineum ergo, H℟Fk, neq; </s>
            <s xml:id="echoid-s11478" xml:space="preserve">maius, neq; </s>
            <s xml:id="echoid-s11479" xml:space="preserve">mi-
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            nus eſt ſpatio reſiduo iam dicto, ergo illi æquale, ſicut triangulus,
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              <note position="right" xlink:label="note-0465-01" xlink:href="note-0465-01a" xml:space="preserve">Elicitur
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              ex prima
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              1. 4.</note>
            HFK, eſt æqualis portioni circuli, cuius baſis eſt circumferentia,
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            CHV, ſed triangulus, HFk, eſt ſexquialter trilinei, H℟FK, ergo
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            talis portio eſt ſexquialtera ſpatij reſidui iam dicti, ergo eſt tripla
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            ſpatij, quod ſpirali, AROV, & </s>
            <s xml:id="echoid-s11480" xml:space="preserve">recta, AV, continetur, quod erat
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            oſtendendum.</s>
            <s xml:id="echoid-s11481" xml:space="preserve"/>
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        <div xml:id="echoid-div1056" type="section" level="1" n="635">
          <head xml:id="echoid-head665" xml:space="preserve">THEOREMA XI. PROPOS. XI.</head>
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            <s xml:id="echoid-s11482" xml:space="preserve">SI ab initio ſpiralis in prima reuolutione ortæ educan-
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            tur rectæ lineæ vtcumque ad ipſam ſpiralem terminã-
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            tes, ſpatia ſub portionibus ſpiralis abſciſsis per eductas
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            verſus initium, erunt vt cubi earundem eductarum.</s>
            <s xml:id="echoid-s11483" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s11484" xml:space="preserve">Sit ſpiralis in prima reuolutione orta, ACDB, ipſa, AB, reuo-
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            luta, & </s>
            <s xml:id="echoid-s11485" xml:space="preserve">ſpiralis initium, A, à quo ad ipſam ſpiralem terminantes
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            ſint eductæ vtcumq; </s>
            <s xml:id="echoid-s11486" xml:space="preserve">AC, AD. </s>
            <s xml:id="echoid-s11487" xml:space="preserve">Dico ſpatium ſub portione ſpira-
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            lis, AXC, & </s>
            <s xml:id="echoid-s11488" xml:space="preserve">educta, AC, ad ſpatium ſub portione ſpiralis, AXCD,
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            & </s>
            <s xml:id="echoid-s11489" xml:space="preserve">educta, AD, eſſe vt cubum, AC, ad cubum, AD. </s>
            <s xml:id="echoid-s11490" xml:space="preserve">Centro igitur,
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            A, interuallis, C, D, ſint deſcripti circuli, CMVN, DGE, & </s>
            <s xml:id="echoid-s11491" xml:space="preserve"/>
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