Archimedes
,
Archimedis De iis qvae vehvntvr in aqva libri dvo
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0065
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DE IIS QVAE VEH. IN AQVA.
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æqualis r ψ: </
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>
<
s
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xml:space
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">& </
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<
s
xml:id
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"
xml:space
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">ducatur ψ r perpendicularis ad b d, quæ
<
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posſit dimidium eius, quod ipſis k r, ψ b, continetur. </
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<
s
xml:id
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xml:space
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">Dico
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lb
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portionem in humidum demiſſam adeo, ut baſis ipſius to-
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ta ſit in humido, ita conſiſtere, ut axis cum ſuperficie humi
<
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/>
di faciat angulum angulo b æqualem. </
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>
<
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xml:id
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xml:space
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">Demittatur enim
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lb
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portio in humidum, ſicuti dictum eſt; </
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>
<
s
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xml:space
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">& </
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>
<
s
xml:id
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echoid-s1568
"
xml:space
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">axis cum humidi
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/>
ſuperficie non faciat angulum æqualẽ ipſi b, ſed primo ma
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/>
iorem: </
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>
<
s
xml:id
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echoid-s1569
"
xml:space
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">ſecta autem ipſa plano per axem, recto ad ſuperfi-
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lb
/>
ciem humidi, ſectio portionis ſit a p o l rectanguli coni ſe-
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/>
ctio; </
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<
s
xml:id
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xml:space
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">ſuperficiei humidi ſectio c i; </
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<
s
xml:id
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xml:space
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">ſitq, axis portionis, & </
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>
<
s
xml:id
="
echoid-s1572
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xml:space
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">ſe
<
lb
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ctionis diameter n o, quæ fecetur in punctis ω t, ut prius. </
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<
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xml:space
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">& </
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<
s
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<
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ducantur y p quidem ipſi ci æquidiſtans, contingensq; </
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<
s
xml:id
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echoid-s1575
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xml:space
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">ſe
<
lb
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ctionem in p; </
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<
s
xml:id
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xml:space
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">m p uero æquidiſtans n o: </
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<
s
xml:id
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xml:space
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">& </
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>
<
s
xml:id
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xml:space
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">p s ad axem
<
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perpendicularis. </
s
>
<
s
xml:id
="
echoid-s1579
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xml:space
="
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">Quoniam igitur axis portionis cum ſu-
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perficie humidi facit angulum maiorem angulo b; </
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<
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xml:space
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">erit & </
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<
s
xml:id
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xml:space
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">
<
lb
/>
angulus s y p angulo b maior. </
s
>
<
s
xml:id
="
echoid-s1582
"
xml:space
="
preserve
">quare quadratum p s ad
<
lb
/>
quadratum s y maiorem habet proportionem, quàm qua
<
lb
/>
dratum ψ e ad quadratum ψ b: </
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>
<
s
xml:id
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xml:space
="
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">& </
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>
<
s
xml:id
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"
xml:space
="
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">propterea _K_ r ad s y ma
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xlink:label
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xlink:href
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xml:space
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">B</
note
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iorem habet, quàm dimidium ipſius κ r ad ψ b. </
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<
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xml:space
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">ergo s y
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minor eſt, quam dupla ψ b; </
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<
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xml:space
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<
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xml:space
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">s o minor, quam ψ b. </
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xml:space
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">quare
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<
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xlink:label
="
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xlink:href
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note-0065-02a
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xml:space
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">C</
note
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s ω maior, quàm r ψ; </
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<
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xml:space
="
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>
<
s
xml:id
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xml:space
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">p h maior, quàm f. </
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>
<
s
xml:id
="
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"
xml:space
="
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">Itaque quoniã
<
lb
/>
portio ad humidum in grauitate eam habet proportionẽ,
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xlink:label
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xml:space
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">D</
note
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quam exceſſus, quo quadratum b d excedit quadratum f q
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ad quadratum b d: </
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>
<
s
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xml:space
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">quam uero proportionem habet por-
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/>
tio ad humidum in grauitate, eandem pars ipſius demerſa
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habet ad totam portionẽ: </
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<
s
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xml:space
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">ſequitur partẽ demerſam ad to
<
lb
/>
tam portionem, eam proportionem habere, quã exceſſus,
<
lb
/>
quo quadratum b d excedit quadratũ f q, ad quadratū b d.
<
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</
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>
<
s
xml:id
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xml:space
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">habebit ergo tota portio ad eam, quæ eſt extra humidum
<
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<
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proportionem eandem, quam quadratum b d ad quadra-
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tum f q. </
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<
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xml:space
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">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:id
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"
xml:space
="
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">ergo p m ipſi f q æ qualis etit. </
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>
<
s
xml:id
="
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"
xml:space
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">demonſtra
<
lb
/>
ta eſt autem p h maior, quàm f: </
s
>
<
s
xml:id
="
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"
xml:space
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">quare m h minor </
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>
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