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

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          <pb o="455" file="0475" n="475" rhead="LIBER VI."/>
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        <div xml:id="echoid-div1069" type="section" level="1" n="641">
          <head xml:id="echoid-head671" xml:space="preserve">THEOREMA XVI. PROPOS. XVI.</head>
          <p>
            <s xml:id="echoid-s11659" xml:space="preserve">SI in ſpirali ex quacunq; </s>
            <s xml:id="echoid-s11660" xml:space="preserve">reuolutione genita ſumatur
              <lb/>
            punctum, quod non ſit initium, nec terminus eiuſdem
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            ſpiralis, & </s>
            <s xml:id="echoid-s11661" xml:space="preserve">iungantur cum puncto, quod eſt initium reuo-
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            lutionis, quo tanquam centro ad diſtantiam ſumpti puncti
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            circulus ſit deſcriptus, huius ſector, vel ſectoris reſiduum,
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            cuius baſis ſit circumferentia inter hoc punctum, & </s>
            <s xml:id="echoid-s11662" xml:space="preserve">princi-
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            pium circulationis ad partes conſequentes incluſa, ad ſpa-
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            tium helicum ab eodem ſectore, vel ſectoris reſiduo, ap-
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            prehenſum, erit vt quadratum ſemidiametri deſcripti cir-
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            culi, ad rectangulum ſub eodem, & </s>
            <s xml:id="echoid-s11663" xml:space="preserve">ſub radio circuli eiuf-
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            dem numeri cum ſpirali vnitate prædicta minoris, vna cũ
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            tertia parte quadrati exceſſus vtriuſq; </s>
            <s xml:id="echoid-s11664" xml:space="preserve">radij.</s>
            <s xml:id="echoid-s11665" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11666" xml:space="preserve">Conſpiciatur antecedentis figura, in qua ſumpto vtcunq; </s>
            <s xml:id="echoid-s11667" xml:space="preserve">pun-
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            cto in ſpirali, GMSIB, quod ſit, I, intelligatur deſcriptus circulus,
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            IRεK. </s>
            <s xml:id="echoid-s11668" xml:space="preserve">Dico igitur ſectorem, vel eius reſiduum, cuius baſis eſt
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            circumferentia, IRεk, ad rectas, IA, Ak, terminata, ad ſpatium
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            ſub ſpiralis portiones, ISMG, & </s>
            <s xml:id="echoid-s11669" xml:space="preserve">rectis, IA, AG, eſſe vt quadratũ,
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            IA, ad rectangulum ſub, IA, AG, vna cum {1/3}. </s>
            <s xml:id="echoid-s11670" xml:space="preserve">quadrati, Gk; </s>
            <s xml:id="echoid-s11671" xml:space="preserve">in ip-
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            ſa enim, QΣ, iam habebus, & </s>
            <s xml:id="echoid-s11672" xml:space="preserve">Σ, æqualem circumferentiæ, CDFB,
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            terminanti ad, C, B, producatur, Φ+, quouſque ſecet ambas, LΠ, L
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            Σ, vt in, {12/ }, {13/ }, & </s>
            <s xml:id="echoid-s11673" xml:space="preserve">quia, &</s>
            <s xml:id="echoid-s11674" xml:space="preserve">Σ, ad, Z, {13/ }, eſt vt, ΣL, ad, L, {13/ }, vel vt, PL, ad,
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              <note position="right" xlink:label="note-0475-01" xlink:href="note-0475-01a" xml:space="preserve">Iuxta 4.
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              Sexti Ele.</note>
            L4, ſiue, BA, ad, AK, ſiue circumferentia, CDFB, ad circumferẽtiam,
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            IRεK, ideò circumferentia, IRεk, erit æqualis ipſi, Z {13/ }, ſi ergo diui-
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            damus, Z, {13/ }, bifariam, & </s>
            <s xml:id="echoid-s11675" xml:space="preserve">factas portiones adhuc bifariam, & </s>
            <s xml:id="echoid-s11676" xml:space="preserve">ſic sẽ-
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            per fiat, iungẽtes diuiſionum pũcta cum, L, & </s>
            <s xml:id="echoid-s11677" xml:space="preserve">per puncta, in quibus
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            iſtę iungẽtes ſecant curuã parabolę, ΖΩ, ductis ipſi, Z {13/ }, parallelis,
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            vt in antecedenti circumſcripſerimus trilineo, LΖΩ, figuram, & </s>
            <s xml:id="echoid-s11678" xml:space="preserve">aliã
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            inſcripſerimus, ex triangulis compoſitam, & </s>
            <s xml:id="echoid-s11679" xml:space="preserve">ſimiliter ſpatio, AIS
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            MGA, figuram ex ſectoribus, vel eorum reſiduis compoſitam cir-
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            cumſcripſerimus, velut in antecedenti (quam quia antecedentis
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            propoſitionis methodo ſimilis eſt, hic explanare mitto) & </s>
            <s xml:id="echoid-s11680" xml:space="preserve">aliam
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            inſcripſerimus, tandem oſtendemus trilineum, LΖΩ, neq; </s>
            <s xml:id="echoid-s11681" xml:space="preserve">maius,
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            neq; </s>
            <s xml:id="echoid-s11682" xml:space="preserve">minus eſſe ſpatio, AISMGA, & </s>
            <s xml:id="echoid-s11683" xml:space="preserve">ideò illi eſſe æquale; </s>
            <s xml:id="echoid-s11684" xml:space="preserve">ſimili-
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            ter oſten demus triangulum, LZ {13/ }, ſectori, IPεK, vel ſectoris reſi-
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              <note position="right" xlink:label="note-0475-02" xlink:href="note-0475-02a" xml:space="preserve">Defin. 12.
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              l. 1.</note>
            duo, æqualem eſſe, nam triangulus, LQΣ, ad triangulum, LZ {13/ </s>
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