Archimedes
,
Archimedis De iis qvae vehvntvr in aqva libri dvo
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DE IIS QVAE VEH. IN AQVA.
"/>
ad ſectionem e f g ex parte e linea l m, eidem a c baſi æquidi-
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stans. </
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<
s
xml:id
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xml:space
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">Sit autem ſectionis a b c, linea b n iuxta quam poſſunt, quæ
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à ſectione ducuntur: </
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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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">ſectionis e f c ſit ipſa f o. </
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>
<
s
xml:id
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echoid-s1863
"
xml:space
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">quoniam igi-
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tur triangula c d b, c f g ſimilia ſunt, erit ut b c ad c f, ita d c
<
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<
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xml:space
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">4. ſexti.</
note
>
ad c g; </
s
>
<
s
xml:id
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xml:space
="
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">& </
s
>
<
s
xml:id
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xml:space
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">b d ad f g. </
s
>
<
s
xml:id
="
echoid-s1866
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xml:space
="
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">rurſus quoniam triangula c k b, c l f etiã
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inter ſe ſunt ſimilia, ut b c ad c f, boc eſt ut b d ad f g, ita erit k c
<
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ad c l; </
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<
s
xml:id
="
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xml:space
="
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">& </
s
>
<
s
xml:id
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xml:space
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">b K ad f l. </
s
>
<
s
xml:id
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xml:space
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">quare K c ad c l, & </
s
>
<
s
xml:id
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xml:space
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">b k ad f l ſunt ut d c
<
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ad c g: </
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<
s
xml:id
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xml:space
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">hoc eſt ut earum duplæ a c ad c e. </
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<
s
xml:id
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xml:space
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">ſed ut b d ad f g, ita d c
<
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<
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xlink:label
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xlink:href
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xml:space
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">15. quin-
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ti.</
note
>
ad c g; </
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<
s
xml:id
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xml:space
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">hoc ẽ a d ad e g: </
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>
<
s
xml:id
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xml:space
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">& </
s
>
<
s
xml:id
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xml:space
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">permutãdo ut b d ad a d, ita f g ad e g.
<
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/>
</
s
>
<
s
xml:id
="
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"
xml:space
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">quadratum autem a d æquale eſt rectangulo d b n ex undecima pri
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mi conicorum. </
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<
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xml:id
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xml:space
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">ergo tres lineæ b d, a d, b n inter ſe ſunt proportio
<
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<
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xml:space
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">17. ſexti.</
note
>
nales. </
s
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<
s
xml:id
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xml:space
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">eadem quoque ratione cum quadratum e g æquale ſit rectan
<
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gulo g f o, tres aliæ lineæ f g, e g, f o, deinceps proportionales
<
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erũt. </
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<
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xml:id
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xml:space
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">& </
s
>
<
s
xml:id
="
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xml:space
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">ut b d ad, a d, ita f g ad e g. </
s
>
<
s
xml:id
="
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xml:space
="
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">quare ut a d ad b n, ita e g
<
lb
/>
ad f o. </
s
>
<
s
xml:id
="
echoid-s1882
"
xml:space
="
preserve
">ex æquali igitur, ut d b ad b n, ita g f ad f o: </
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>
<
s
xml:id
="
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xml:space
="
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">& </
s
>
<
s
xml:id
="
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xml:space
="
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">permu-
<
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tando ut d b ad g f, ita b n ad f o. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">ut autem d b ad g f, ita b k
<
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ad f l. </
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>
<
s
xml:id
="
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"
xml:space
="
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">ergo b k ad f l, ut b n ad f o: </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s1888
"
xml:space
="
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">permutando, ut b k ad
<
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/>
bn, ita f l ad f o. </
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>
<
s
xml:id
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xml:space
="
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">Rurſus quoniá quadratú h K æquale eſt rectan
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xlink:label
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xlink:href
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xml:space
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">11. primi
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conicorũ</
note
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gulo k b n: </
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<
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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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">quadratum m l rectangulo l f o æquale: </
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<
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xml:id
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xml:space
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">erunt tres
<
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lineæ b k, k h, b n proportionales: </
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<
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xml:id
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xml:space
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">itémq; </
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<
s
xml:id
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xml:space
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">proportionales inter ſe
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f l, l m, f o. </
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<
s
xml:id
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xml:space
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">quare ut linea b K ad lineam b n, ita quadratum b K
<
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<
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="
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xlink:label
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xlink:href
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xml:space
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">cor. 20. ſe
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xti.</
note
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ad quadratum h k: </
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<
s
xml:id
="
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xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s1897
"
xml:space
="
preserve
">ut linea f l ad ipſam f o, ita quadratú f l
<
lb
/>
ad quadratum l m. </
s
>
<
s
xml:id
="
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xml:space
="
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">Itaque quoniam, ut b K ad b n, ita eſt f l ad
<
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f o; </
s
>
<
s
xml:id
="
echoid-s1899
"
xml:space
="
preserve
">erit ut quadratum b K ad quadratum k h, ita quadratum f l
<
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ad l m quadratum. </
s
>
<
s
xml:id
="
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xml:space
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">ergo ut linea b k, ad lineam K h, ita linea f l
<
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<
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position
="
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xlink:label
="
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xlink:href
="
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xml:space
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">22. ſexti</
note
>
ad ipsã lm: </
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<
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xml:id
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echoid-s1901
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xml:space
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">& </
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>
<
s
xml:id
="
echoid-s1902
"
xml:space
="
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">permutãdo ut b k ad f l, ita k h ad lm. </
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>
<
s
xml:id
="
echoid-s1903
"
xml:space
="
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">ſed b k ad
<
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/>
f l erat ut k c ad c l. </
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<
s
xml:id
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xml:space
="
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">ergo k h ad lm, ut K c ad c l. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">quare ex eo
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dem lemmate patet lineam h c, & </
s
>
<
s
xml:id
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xml:space
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">per m punctum tranſire. </
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>
<
s
xml:id
="
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xml:space
="
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">ut igi-
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tur K c ad c l: </
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>
<
s
xml:id
="
echoid-s1908
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xml:space
="
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">hoc eſt ut a c ad c e, ita h c ad c m; </
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>
<
s
xml:id
="
echoid-s1909
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xml:space
="
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">hoc eſt ad eam
<
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ipſius partem, quæ inter c, & </
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<
s
xml:id
="
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xml:space
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">e g c ſectionem interyeitur. </
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>
<
s
xml:id
="
echoid-s1911
"
xml:space
="
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">ſimiliter
<
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demonſtrabimus idem contingere in alijs lineis, quæ à puncto c ad
<
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/>
a b c ſectionem perducuntur. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">At uero b c ad e f eandern propor-
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tionem habere, liquido apparet; </
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>
<
s
xml:id
="
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xml:space
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">nam b c ad c f, eſt ut d c ad c g;
<
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</
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<
s
xml:id
="
echoid-s1914
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xml:space
="
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">uidelicet ut earum duplæ, a c ad c e.</
s
>
<
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="
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xml:space
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