Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[61.] ALEXANDRO FARNESIO CARDINALI AMPLISSIMO ET OPTIMO.
[62.] FEDERICI COMMANDINI VRBINATIS LIBER DE CENTRO GRAVITATIS SOLIDORVM. DIFFINITIONES.
[63.] PETITIONES.
[64.] THEOREMA I. PROPOSITIO I.
[65.] THEOREMA II. PROPOSITIO II.
[66.] THE OREMA III. PROPOSITIO III.
[67.] THE OREMA IIII. PROPOSITIO IIII.
[68.] ALITER.
[69.] THEOREMA V. PROPOSITIO V.
[70.] COROLLARIVM.
[71.] THEOREMA VI. PROPOSITIO VI.
[72.] THE OREMA VII. PROPOSITIO VII.
[73.] THE OREMA VIII. PROPOSITIO VIII.
[74.] THE OREMA IX. PROPOSITIO IX.
[75.] PROBLEMA I. PROPOSITIO X.
[76.] PROBLEMA II. PROPOSITIO XI.
[77.] PROBLEMA III. PROPOSITIO XII.
[78.] PROBLEMA IIII. PROPOSITIO XIII.
[79.] THEOREMA X. PROPOSITIO XIIII.
[80.] THE OREMA XI. PROPOSITIO XV.
[81.] THE OREMA XII. PROPOSITIO XVI.
[82.] THE OREMA XIII. PROPOSITIO XVII.
[83.] THEOREMA XIIII. PROPOSITIO XVIII.
[84.] THEOREMA XV. PROPOSITIO XIX.
[85.] THE OREMA XVI. PROPOSITIO XX.
[86.] THEOREMA XVII. PROPOSITIO XXI.
[87.] THE OREMA XVIII. PROPOSITIO XXII.
[88.] THEOREMA XIX. PROPOSITIO XXIII.
[89.] PROBLEMA V. PROPOSITIO XXIIII.
[90.] THEOREMA XX. PROPOSITIO XXV.
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            <s xml:id="echoid-s3422" xml:space="preserve">
              <pb o="12" file="0135" n="135" rhead="DE CENTRO GRA VIT. SOLID."/>
            Itaque ſolidi parallelepipedi y γ centrum grauitatis eſt in
              <lb/>
            linea δ: </s>
            <s xml:id="echoid-s3423" xml:space="preserve">ſolidi u β centrum eſt in linea ε η: </s>
            <s xml:id="echoid-s3424" xml:space="preserve">& </s>
            <s xml:id="echoid-s3425" xml:space="preserve">ſolidi s z in li
              <lb/>
            nea η m, quæ quidem lineæ axes ſunt, cum planorum oppo
              <lb/>
            ſitorum centra coniungant. </s>
            <s xml:id="echoid-s3426" xml:space="preserve">ergo magnitudinis ex his ſoli
              <lb/>
            dis compoſitæ centrum grauitatis eſt in linea δ m, quod ſit
              <lb/>
            θ; </s>
            <s xml:id="echoid-s3427" xml:space="preserve">& </s>
            <s xml:id="echoid-s3428" xml:space="preserve">iuncta θ o producatur: </s>
            <s xml:id="echoid-s3429" xml:space="preserve">à puncto autem h ducatur h μ
              <lb/>
            ipſi m κ æquidiſtans, quæ cum θ o in μ conueniat. </s>
            <s xml:id="echoid-s3430" xml:space="preserve">triangu
              <lb/>
            lum igitur g h κ ad omnia triangula g z r, r β t, t γ x, x δ k,
              <lb/>
            κ δ y, y u, u s, s α h eandem habet proportionem, quam h m
              <lb/>
            ad m q; </s>
            <s xml:id="echoid-s3431" xml:space="preserve">hoc eſt, quam μ θ ad θ λ: </s>
            <s xml:id="echoid-s3432" xml:space="preserve">nam ſi h m, μ θ produci in
              <lb/>
            telligantur, quouſque coeant; </s>
            <s xml:id="echoid-s3433" xml:space="preserve">erit ob linearum q y, m k æ-
              <lb/>
            quidiſtantiam, ut h q ad q m, ita μ λ ad ad λ θ: </s>
            <s xml:id="echoid-s3434" xml:space="preserve">& </s>
            <s xml:id="echoid-s3435" xml:space="preserve">componen
              <lb/>
            do, ut h m ad m q, ita μ θ ad θ λ. </s>
            <s xml:id="echoid-s3436" xml:space="preserve">linea uero θ o maior eſt,
              <lb/>
            quàm θ λ: </s>
            <s xml:id="echoid-s3437" xml:space="preserve">habebit igitur μ θ ad θ λ maiorem proportio-
              <lb/>
              <note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">8. quinti.</note>
            nem, quàm ad θ o. </s>
            <s xml:id="echoid-s3438" xml:space="preserve">quare triangulum etiam g h k ad omnia
              <lb/>
            iam dicta triangula maiorem proportionẽ habebit, quàm
              <lb/>
            μ θ ad θ o. </s>
            <s xml:id="echoid-s3439" xml:space="preserve">ſed ut triangulũ g h k ad omnia triangula, ita to-
              <lb/>
            tũ priſma a f ad omnia priſmata g z r, r β t, t γ x, x δ k, k δ y,
              <lb/>
            y u, u s, s α h: </s>
            <s xml:id="echoid-s3440" xml:space="preserve">quoniam enim ſolida parallelepipeda æque al
              <lb/>
            ta, eandem inter ſe proportionem habent, quam baſes; </s>
            <s xml:id="echoid-s3441" xml:space="preserve">ut
              <lb/>
            ex trigeſimaſecunda undecimi elementorum conſtat. </s>
            <s xml:id="echoid-s3442" xml:space="preserve">ſunt
              <lb/>
              <note position="right" xlink:label="note-0135-02" xlink:href="note-0135-02a" xml:space="preserve">28. unde
                <lb/>
              cimi</note>
            autem ſolida parallelepipeda priſmatum triangulares ba-
              <lb/>
            ſes habentium dupla: </s>
            <s xml:id="echoid-s3443" xml:space="preserve">ſequitur, ut etiam huiuſmodi priſ-
              <lb/>
              <note position="right" xlink:label="note-0135-03" xlink:href="note-0135-03a" xml:space="preserve">15. quinti</note>
            matainter ſe ſint, ſicut eorum baſes. </s>
            <s xml:id="echoid-s3444" xml:space="preserve">ergo totum priſma ad
              <lb/>
            omnia priſmata maiorem proportionem habet, quam μ θ
              <lb/>
            ad θ o: </s>
            <s xml:id="echoid-s3445" xml:space="preserve">& </s>
            <s xml:id="echoid-s3446" xml:space="preserve">diuidendo ſolida parallelepipeda y γ, u β, s z ad o-
              <lb/>
              <note position="right" xlink:label="note-0135-04" xlink:href="note-0135-04a" xml:space="preserve">19. quinti
                <lb/>
              apud Cã
                <lb/>
              panum.</note>
            mnia prifmata proportionem habent maiorem, quàm μ o
              <lb/>
            ad o θ. </s>
            <s xml:id="echoid-s3447" xml:space="preserve">fiat @ o ad o θ, ut folida parallelepipeda y γ, u β, s z ad
              <lb/>
            omnia priſmata. </s>
            <s xml:id="echoid-s3448" xml:space="preserve">Itaque cum à priſmate a f, cuius cẽtrum
              <lb/>
            grauitatis eſt o, auferatur magnitudo ex ſolidis parallelepi
              <lb/>
            pedis y γ, u β, s z conſtans: </s>
            <s xml:id="echoid-s3449" xml:space="preserve">atque ipfius grauitatis centrum
              <lb/>
            ſit θ: </s>
            <s xml:id="echoid-s3450" xml:space="preserve">reliquæ magnitudinis, quæ ex omnibus priſmatibus
              <lb/>
            conſtat, grauitatis centrum erit in linea θ o producta: </s>
            <s xml:id="echoid-s3451" xml:space="preserve">& </s>
            <s xml:id="echoid-s3452" xml:space="preserve">
              <lb/>
            in puncto ν, ex o ctaua propoſitione eiuſdem libri </s>
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