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

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        <div xml:id="echoid-div1044" type="section" level="1" n="629">
          <p>
            <s xml:id="echoid-s11356" xml:space="preserve">
              <pb o="439" file="0459" n="459" rhead="LIBER V."/>
            circumferentiæ circuli, MSE, erunt ad omnes circumferentias fi-
              <lb/>
            guræ ſpirali, AIE, recta, AE, & </s>
            <s xml:id="echoid-s11357" xml:space="preserve">circumferentia, MSE, concluſæ,
              <lb/>
            & </s>
            <s xml:id="echoid-s11358" xml:space="preserve">per conuerſionem rationis triangulum, ORQ, vel, OZR, ad fi-
              <lb/>
            guram, OGR, erit vt omnes circumferentiæ circuli, MSE, ad om-
              <lb/>
            nes circumferentias ſpatij helici, AIE, ideſt vt circulus ad ſpatium,
              <lb/>
            AIE, (quia curua, AIE, eſt talis conditionis, qualem poſtulat Prop.
              <lb/>
            </s>
            <s xml:id="echoid-s11359" xml:space="preserve">
              <note position="right" xlink:label="note-0459-01" xlink:href="note-0459-01a" xml:space="preserve">1. l. 4.</note>
            6. </s>
            <s xml:id="echoid-s11360" xml:space="preserve">vt elicitur ex Prop. </s>
            <s xml:id="echoid-s11361" xml:space="preserve">7. </s>
            <s xml:id="echoid-s11362" xml:space="preserve">huius) cum verò ſemiparabola, OGRZ,
              <lb/>
            ſit ſexquitertia trianguli, OZR, vnde diuidendo figura, OGR, ſit
              <lb/>
            tertia pars trianguli, OZR, ideò, & </s>
            <s xml:id="echoid-s11363" xml:space="preserve">ſpatium helicum, AIE, tertia
              <lb/>
            pars erit circuli, MSE, quod demonſtrare oportebat.</s>
            <s xml:id="echoid-s11364" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1046" type="section" level="1" n="630">
          <head xml:id="echoid-head660" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s11365" xml:space="preserve">_H_Vcuſq; </s>
            <s xml:id="echoid-s11366" xml:space="preserve">per methodum indiuiſibilium etiam in boc Libro libuit
              <lb/>
            procedere, vt innoteſceret nos poſſe, quæ Archimedes oſtendit
              <lb/>
            Lib. </s>
            <s xml:id="echoid-s11367" xml:space="preserve">de Spiralibus, circa ſpatiorum menſuram, etiam tali artificio de-
              <lb/>
            monſtrare, etenim ſi quis hoc attentauerit circa ſequentes Propoſitio-
              <lb/>
            nes, idipſum obtineri poſſe facilè animaduertet, veruntamen hoc ar-
              <lb/>
            bitrio, ac iudicio Lectoris relinquendo, placuit etiam ſtylo veteri,
              <lb/>
            aliter tamen ab Archimede, eaſdem propoſitiones demonſtrare.</s>
            <s xml:id="echoid-s11368" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1047" type="section" level="1" n="631">
          <head xml:id="echoid-head661" style="it" xml:space="preserve">Præfatæ Propoſ. alia demonſtratio.</head>
          <p>
            <s xml:id="echoid-s11369" xml:space="preserve">SIt alia ſpiralis ex prima reuolutione orta, ASRMB, AB, verò
              <lb/>
            initium reuolutionis, & </s>
            <s xml:id="echoid-s11370" xml:space="preserve">centro, A, interuallo, AB, ſit primus
              <lb/>
            circulus deſcriptus, ECDB, deinde exponatur triangulus, FHG,
              <lb/>
            rectum habens angulum ad, G, cuius latus, FG, ſit æquale ipſi, A
              <lb/>
            B, & </s>
            <s xml:id="echoid-s11371" xml:space="preserve">HG, circumferentiæ, ECDB, erit ergo triangulus, FHG, æ-
              <lb/>
            qualis circulo, ECDB, intelligatur deinde in eiuſdem trianguli
              <lb/>
              <note position="right" xlink:label="note-0459-02" xlink:href="note-0459-02a" xml:space="preserve">z. huius.</note>
            plano tranſire parabolam, HLF, cuius vertex ſit, F, &</s>
            <s xml:id="echoid-s11372" xml:space="preserve">, HG, pa-
              <lb/>
              <note position="right" xlink:label="note-0459-03" xlink:href="note-0459-03a" xml:space="preserve">20. l. 4.</note>
            rallela eiuſdem axi, ad quemipſa, GF, ſit ordinatim applicata, quę
              <lb/>
            tanget ſectionem in puncto, F. </s>
            <s xml:id="echoid-s11373" xml:space="preserve">Dico igitur, FLHG, trilineum
              <lb/>
              <note position="right" xlink:label="note-0459-04" xlink:href="note-0459-04a" xml:space="preserve">17. Primi
                <lb/>
              Conic.</note>
            æquari ſpatio reſiduo, dempto à circulo, ECDB, ſpatio helico ſub
              <lb/>
            ſpirali, ASRMB, &</s>
            <s xml:id="echoid-s11374" xml:space="preserve">, AB, ſi enim non eſt illi æquale, erit eodem,
              <lb/>
              <note position="right" xlink:label="note-0459-05" xlink:href="note-0459-05a" xml:space="preserve">Defi. 3.
                <lb/>
              huius.</note>
            vel maius, vel minus, ſit primò maius quantitate ſpatij, quod vo-
              <lb/>
            cetur, Ω, rurſus diuidatur, HG, bifariam in, Π, & </s>
            <s xml:id="echoid-s11375" xml:space="preserve">iungantur, FΠ,
              <lb/>
            & </s>
            <s xml:id="echoid-s11376" xml:space="preserve">ſic ipſæ, AΠ, ΠG, diuidantur bifariam in, P, Γ, & </s>
            <s xml:id="echoid-s11377" xml:space="preserve">iungantur,
              <lb/>
            PF, ΓF, ſicque ſemper fiat donec deuentum ſit, vt ad triangulum,
              <lb/>
            FΓG, quod ſit minus ſpatio, Ω, deueniemus autem, nam à ma-
              <lb/>
            gnitudine propoſita, & </s>
            <s xml:id="echoid-s11378" xml:space="preserve">his, quæ relinquuntur, ſemper aufertur
              <lb/>
              <note position="right" xlink:label="note-0459-06" xlink:href="note-0459-06a" xml:space="preserve">1. Decimè
                <lb/>
              Elem.</note>
            dimidium, ſecent autem iungentes, F, cum diuiſionum punctis </s>
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