Archimedes
,
Archimedis De iis qvae vehvntvr in aqva libri dvo
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rhead
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DE IIS QVAE VEH. IN AQVA.
"/>
itêmq; </
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<
s
xml:id
="
echoid-s1937
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xml:space
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">quadratum c q æquale rectangulo q u y, hoc eſt ſectionum
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h s c, m u c lineas s x, u y, eas eſſe, iuxta quas poſſunt, quæ à ſectio-
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ne ad diametrum ducuntur. </
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<
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xml:id
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echoid-s1938
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xml:space
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preserve
">ſed cú triangula c p r, c q t ſimilia ſint,
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habebit c r ad c p eandem proportionem, quam c t ad c q: </
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<
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echoid-s1939
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xml:space
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">& </
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<
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">id-
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<
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">22. fexti</
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circo quadratum c r ad quadratum c p eandem habebit, quam
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quadratum c t ad quadratum c q. </
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<
s
xml:id
="
echoid-s1941
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">ergo & </
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>
<
s
xml:id
="
echoid-s1942
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xml:space
="
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">linea b n, ad lineam
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/>
ſ x ita erit, ut linea fo ad ipſam u y. </
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>
<
s
xml:id
="
echoid-s1943
"
xml:space
="
preserve
">erat autem b c ad c m, ut a c
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ad c e. </
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<
s
xml:id
="
echoid-s1944
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">quare & </
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<
s
xml:id
="
echoid-s1945
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="
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">earum dimidiæ c p ad c q, ut a d ad e g: </
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<
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="
echoid-s1946
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xml:space
="
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">& </
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<
s
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<
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permutando c p ad a d, ut c q ad e g. </
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>
<
s
xml:id
="
echoid-s1948
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xml:space
="
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">Sed oſtenſum est a d ad b n
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ita eſſe, ut e g ad f o: </
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<
s
xml:id
="
echoid-s1949
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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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">b n ad s x, ut f o ad u y. </
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>
<
s
xml:id
="
echoid-s1951
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xml:space
="
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">ergo ex
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æquali c p ad ſ x erit, ut c q ad u y. </
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<
s
xml:id
="
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xml:space
="
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">Quòd cum quadratú c p æqua
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le ſit rectangulo p s x & </
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<
s
xml:id
="
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xml:space
="
preserve
">quadratum c q rectangulo q u y, erunt
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tres lineæ ſ p, p c, ſ x proportionales; </
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>
<
s
xml:id
="
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xml:space
="
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">itemq; </
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>
<
s
xml:id
="
echoid-s1955
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xml:space
="
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">proportionales ip-
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ſæ u q, q c, u y. </
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<
s
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">quare & </
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<
s
xml:id
="
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xml:space
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">ſ p ad p c, ut u q ad q c: </
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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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">ut p c ad
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c h, ita q c ad c m. </
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<
s
xml:id
="
echoid-s1960
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xml:space
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">ex æquali igitur ut portionis h ſ c diameter ſ p
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ad eius baſim c h, ita portionis m u s diameter u q ad baſim c m.
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</
s
>
<
s
xml:id
="
echoid-s1961
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xml:space
="
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">& </
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<
s
xml:id
="
echoid-s1962
"
xml:space
="
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">anguli, quos diametri cum baſibus continent, ſunt æquales, quòd
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lineæ ſ p, u q ſibi ipſis æquidiſtent, ergo & </
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>
<
s
xml:id
="
echoid-s1963
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xml:space
="
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">portiones h ſ c, m u c
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inter ſe ſimiles erunt. </
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>
<
s
xml:id
="
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="
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">id quod demonstrandum proponebatur.</
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="
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head
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">LEMMA IIII.</
head
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<
s
xml:id
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xml:space
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">Sint duæ lineæ a b, c d, quæ ſecentur in punctis e f,
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ita ut quam proportionem habet a e ad e b, habeat c f
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ad f d: </
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>
<
s
xml:id
="
echoid-s1967
"
xml:space
="
preserve
">rurſus ſecentur in aliis duobus punctis g h; </
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<
s
xml:id
="
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="
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">& </
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>
<
s
xml:id
="
echoid-s1969
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habeat c h ad h d eandem proportionem, quam a g ad
<
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/>
g b. </
s
>
<
s
xml:id
="
echoid-s1970
"
xml:space
="
preserve
">Dico c f ad f h ita eſſe, ut a e ad e g.</
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>
<
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xml:id
="
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xml:space
="
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"/>
</
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<
s
xml:id
="
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<
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style
="
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">Q_voniam_</
emph
>
enim ut a e ad e b, ita c f ad f d, erit componen
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do ut a b ad e b, ita c d ad f d. </
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>
<
s
xml:id
="
echoid-s1973
"
xml:space
="
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">Rurſus cum ſit ut a g ad g b, ita
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c h ad h d; </
s
>
<
s
xml:id
="
echoid-s1974
"
xml:space
="
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">componendo, conuertendoq; </
s
>
<
s
xml:id
="
echoid-s1975
"
xml:space
="
preserve
">ut g b ad a b, ita erit h d
<
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/>
ad c d. </
s
>
<
s
xml:id
="
echoid-s1976
"
xml:space
="
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">ergo ex æquali, conuertendoq; </
s
>
<
s
xml:id
="
echoid-s1977
"
xml:space
="
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">ut e b ad g b, ita f d ad h d:</
s
>
<
s
xml:id
="
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xml:space
="
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