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 IIS QVAE VEH. IN AQVA.
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              <pb o="27" file="0065" n="65" rhead="DE IIS QVAE VEH. IN AQVA."/>
            æqualis r ψ: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ducatur ψ r perpendicularis ad b d, quæ
              <lb/>
            posſit dimidium eius, quod ipſis k r, ψ b, continetur. </s>
            <s xml:space="preserve">Dico
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            portionem in humidum demiſſam adeo, ut baſis ipſius to-
              <lb/>
            ta ſit in humido, ita conſiſtere, ut axis cum ſuperficie humi
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            di faciat angulum angulo b æqualem. </s>
            <s xml:space="preserve">Demittatur enim
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            portio in humidum, ſicuti dictum eſt; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">axis cum humidi
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            ſuperficie non faciat angulum æqualẽ ipſi b, ſed primo ma
              <lb/>
            iorem: </s>
            <s xml:space="preserve">ſecta autem ipſa plano per axem, recto ad ſuperfi-
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            ciem humidi, ſectio portionis ſit a p o l rectanguli coni ſe-
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            ctio; </s>
            <s xml:space="preserve">ſuperficiei humidi ſectio c i; </s>
            <s xml:space="preserve">ſitq, axis portionis, & </s>
            <s xml:space="preserve">ſe
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            ctionis diameter n o, quæ fecetur in punctis ω t, ut prius. </s>
            <s xml:space="preserve">& </s>
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            ducantur y p quidem ipſi ci æquidiſtans, contingensq; </s>
            <s xml:space="preserve">ſe
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            ctionem in p; </s>
            <s xml:space="preserve">m p uero æquidiſtans n o: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">p s ad axem
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            perpendicularis. </s>
            <s xml:space="preserve">Quoniam igitur axis portionis cum ſu-
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            perficie humidi facit angulum maiorem angulo b; </s>
            <s xml:space="preserve">erit & </s>
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            angulus s y p angulo b maior. </s>
            <s xml:space="preserve">quare quadratum p s ad
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            quadratum s y maiorem habet proportionem, quàm qua
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            dratum ψ e ad quadratum ψ b: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">propterea _K_ r ad s y ma
              <lb/>
              <anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/>
            iorem habet, quàm dimidium ipſius κ r ad ψ b. </s>
            <s xml:space="preserve">ergo s y
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            minor eſt, quam dupla ψ b; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">s o minor, quam ψ b. </s>
            <s xml:space="preserve">quare
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              <anchor type="note" xlink:label="note-0065-02a" xlink:href="note-0065-02"/>
            s ω maior, quàm r ψ; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">p h maior, quàm f. </s>
            <s xml:space="preserve">Itaque quoniã
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            portio ad humidum in grauitate eam habet proportionẽ,
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              <anchor type="note" xlink:label="note-0065-03a" xlink:href="note-0065-03"/>
            quam exceſſus, quo quadratum b d excedit quadratum f q
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            ad quadratum b d: </s>
            <s xml:space="preserve">quam uero proportionem habet por-
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            tio ad humidum in grauitate, eandem pars ipſius demerſa
              <lb/>
            habet ad totam portionẽ: </s>
            <s xml:space="preserve">ſequitur partẽ demerſam ad to
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            tam portionem, eam proportionem habere, quã exceſſus,
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            quo quadratum b d excedit quadratũ f q, ad quadratū b d.
              <lb/>
            </s>
            <s xml:space="preserve">habebit ergo tota portio ad eam, quæ eſt extra humidum
              <lb/>
              <anchor type="note" xlink:label="note-0065-04a" xlink:href="note-0065-04"/>
            proportionem eandem, quam quadratum b d ad quadra-
              <lb/>
            tum f q. </s>
            <s xml:space="preserve">Sed quam proportionem habet tota portio ad eã,
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            quæ eſt extra humidum, eandem habet quadratum n o ad
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            quadratum p m. </s>
            <s xml:space="preserve">ergo p m ipſi f q æ qualis etit. </s>
            <s xml:space="preserve">demonſtra
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            ta eſt autem p h maior, quàm f: </s>
            <s xml:space="preserve">quare m h minor erit,</s>
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