Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

Table of contents

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[41. COMMENTARIVS.]
[42. LEMMA I.]
[43. LEMMA II.]
[44. LEMMA III.]
[45. LEMMA IIII.]
[46. LEMMA V.]
[47. LEMMA VI.]
[48. II.]
[49. III.]
[50. IIII.]
[51. V.]
[52. DEMONSTRATIO SECVNDAE PARTIS.]
[53. COMMENTARIVS.]
[54. DEMONSTRATIO TERTIAE PARTIS.]
[55. COMMENTARIVS.]
[56. DEMONSTRATIO QVARTAE PARTIS.]
[57. DEMONSTRATIO QVINT AE PARTIS.]
[58. FINIS LIBRORVM ARCHIMEDIS DE IIS, QVAE IN AQVA VEHVNTVR.]
[59. FEDERICI COMMANDINI VRBINATIS LIBER DE CENTRO GRAVITATIS SOLIDORV M.]
[60. CVM PRIVILEGIO IN ANNOS X. BONONIAE, Ex Officina Alexandri Benacii. M D LXV.]
[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.]
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FED. COMMANDINI
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            habeat circulus, uel ellipſis g h ad aliud ſpacium, in quo u:
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            </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">in circulo, uel ellipſi plane deſcribatur rectilinea figura,
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            ita ut tãdem relinquãtur portiones minores ſpacio u, quæ
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            ſit o p g q r s h t: </s>
            <s xml:space="preserve">deſcriptaq; </s>
            <s xml:space="preserve">ſimili figura in oppoſitis pla-
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            nis c d, f e, per lineas ſibi ipſis reſpondentes plana ducãtur. </s>
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              <lb/>
            Itaque cylindrus, uel cylindri portio diuiditur in priſma,
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            cuius quidem baſis eſt figura rectilinea iam dicta, centrum
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            que grauitatis punctum K: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">in multa ſolida, quæ pro baſi
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            bus habent relictas portiones, quas nos ſolidas portiones
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            appellabimus. </s>
            <s xml:space="preserve">cum igitur portiones ſint minores ſpacio
              <lb/>
            u, circulus, uel ellipſis g h ad portiones maiorem propor-
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            tionem habebit, quàm linea m k ad K l. </s>
            <s xml:space="preserve">fiat n k ad K l, ut
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            circulus uel ellipſis g h ad ipſas portiones. </s>
            <s xml:space="preserve">Sed ut circulus
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            uel ellipſis g h ad figuram rectilineam in ipſa deſcri-
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            ptam, ita eſt cylindrus uel cylindri portio c e ad priſma,
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            quod rectilineam figuram pro baſi habet, & </s>
            <s xml:space="preserve">altitudinem
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            æqualem; </s>
            <s xml:space="preserve">id, quod infra demonſtrabitur, ergo per conuer
              <lb/>
            ſionem rationis, ut circulus, uel ellipſis g h ad portiones re
              <lb/>
            lictas, ita cylindrus, uel cylindri portio c e ad ſolidas por-
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            tiones, quare cylindrus uel cylindri portio ad ſolidas por-
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            tiones eandem proportionem habet, quam linea n k a d _k_
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            & </s>
            <s xml:space="preserve">diuidendo priſma, cuius baſis eſt rectilinea figura ad ſo-
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            lidas portiones eandem proportionem habet, quam n lad
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            1 _k_. </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">quoniam a cylindro uel cylindri portione, cuius gra-
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            uitatis centrum eſt l, aufertur priſma baſim habens rectili-
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            neam figurã, cuius centrũ grauitatis eſt _K_: </s>
            <s xml:space="preserve">reſiduæ magnitu
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            dinis ex ſolidis portionibus cõpoſitæ grauitatis cẽtrũ erit
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            in linea k l protracta, & </s>
            <s xml:space="preserve">in puncto n; </s>
            <s xml:space="preserve">quod eſt abſurdū. </s>
            <s xml:space="preserve">relin
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            quitur ergo, ut cẽtrum grauitatis cylindri; </s>
            <s xml:space="preserve">uel cylin dri por
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            tionis ſit punctũ k. </s>
            <s xml:space="preserve">quæ omnia demonſtrãda propoſuimus.</s>
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              <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0139-01"/>
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            <note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">4. huius.</note>
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          <p>
            <s xml:space="preserve">At uero cylindrum, uel cylindri portionẽ ce
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            ad priſma, cuius baſis eſt rectilinea figura in ſpa-
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            cio g h deſcripta, & </s>
            <s xml:space="preserve">altitudo æqualis; </s>
            <s xml:space="preserve">eandem ha-</s>
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