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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DE CENTRO GRAVIT. SOLID.
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          <p>
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              <pb o="30" file="0171" n="171" rhead="DE CENTRO GRAVIT. SOLID."/>
            pra demonſtratum eſt, ita eſſe cylindrum, uel cylindri por-
              <lb/>
              <anchor type="note" xlink:label="note-0171-01a" xlink:href="note-0171-01"/>
            tionem ad priſina, cuius baſis rectilinea figura, & </s>
            <s xml:space="preserve">æqua-
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            lis altitudo. </s>
            <s xml:space="preserve">ergo per conuerſionem rationis, ut circulus,
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            uel ellipſis ad portiones, ita conus, uel coni portio ad por-
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            tiones ſolidas. </s>
            <s xml:space="preserve">quare conus uel coni portio ad portiones
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            ſolidas maiorem habet proportionem, quam g e ad e f: </s>
            <s xml:space="preserve">& </s>
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              <lb/>
            diuidendo, pyramis ad portiones ſolidas maiorem pro-
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            portionem habet, quam g f ad f e. </s>
            <s xml:space="preserve">ſiat igitur q f ad f e
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            ut pyramis ad dictas portiones. </s>
            <s xml:space="preserve">Itaque quoniam à cono
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            uel coni portione, cuius grauitatis centrum eſt f, aufer-
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            tur pyramis, cuius centrum e; </s>
            <s xml:space="preserve">reliquæ magnitudinis,
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            quæ ex ſolidis portionibus conſtat, centrum grauitatis
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            erit in linea e f protracta, & </s>
            <s xml:space="preserve">in puncto q. </s>
            <s xml:space="preserve">quod fieri
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            non poteft: </s>
            <s xml:space="preserve">eſt enim centrum grauitatis intra. </s>
            <s xml:space="preserve">Conſtat
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            igitur coni, uel coni portionis grauitatis centrum eſſe pun
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            ctum e. </s>
            <s xml:space="preserve">quæ omnia demonſtrare oportebat.</s>
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            <figure xlink:label="fig-0170-01" xlink:href="fig-0170-01a">
              <image file="0170-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0170-01"/>
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            <note position="right" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve">8. huius</note>
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        <div type="section" level="1" n="88">
          <head xml:space="preserve">THEOREMA XIX. PROPOSITIO XXIII.</head>
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              <emph style="sc">Qvodlibet</emph>
            fruſtum à pyramide, quæ
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            triangularem baſim habeat, abſciſſum, diuiditur
              <lb/>
            in tres pyramides proportionales, in ea proportio
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            ne, quæ eſt lateris maioris baſis ad latus minoris
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            ipſi reſpondens.</s>
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          <p>
            <s xml:space="preserve">Hoc demonſtrauit Leonardus Piſanus in libro, qui de-
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            praxi geometriæ inſcribitur. </s>
            <s xml:space="preserve">Sed quoniam is adhucim-
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            preſſus non eſt, nos ipſius demonſtrationem breuíter
              <lb/>
            perſtringemus, rem ipſam ſecuti, non uerba. </s>
            <s xml:space="preserve">Sit fru-
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            ſtum pyramidis a b c d e f, cuíus maior baſis triangulum
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            a b c, minor d e f: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">iunctis a e, e c, c d, per line-
              <lb/>
            as a e, e c ducatur planum ſecans fruſtum: </s>
            <s xml:space="preserve">itemque per
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            lineas e c, c d; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">per c d, d a alia plana ducantur, quæ,
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            diuident fruſtum in tres pyramides a b c e, a d c e, d e f c.</s>
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          </p>
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