Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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[11. PROPOSITIO IIII.]
[12. PROPOSITIO V.]
[13. PROPOSITIO VI.]
[14. PROPOSITIO VII.]
[15. POSITIO II.]
[16. COMMENTARIVS.]
[17. PROPOSITIO VIII.]
[18. COMMENTARIVS.]
[19. PROPOSITIO IX.]
[20. COMMENTARIVS.]
[21. ARCHIMEDIS DE IIS QVAE VEHVNTVR IN AQVA LIBER SECVNDVS. CVM COMMENTARIIS FEDERICI COMMANDINI VRBINATIS. PROPOSITIO I.]
[22. PROPOSITIO II.]
[23. COMMENTARIVS.]
[24. PROPOSITIO III.]
[25. PROPOSITIO IIII.]
[26. COMMENTARIVS.]
[27. PROPOSITIO V.]
[28. COMMENTARIVS.]
[29. PROPOSITIO VI.]
[30. COMMENTARIVS.]
[31. LEMMAI.]
[32. LEMMA II.]
[33. LEMMA III.]
[34. LEMMA IIII.]
[35. PROPOSITIO VII.]
[36. PROPOSITIO VIII.]
[37. COMMENTARIVS.]
[38. PROPOSITIO IX.]
[39. COMMENTARIVS.]
[40. PROPOSITIO X.]
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DE IIS QVAE VEH. IN AQVA.
eam proportionem babebit, quam a f ad a e. Sed & eandem habet
a s ad a r.
quare a s ipſi a x eſt æqualis, pars toti, quod fieri non
9. quintipoteſt.
Idem abſurdum ſequetur, ſi ponamus punctum t cadere ul-
tra lineam a c.
neceſſarium igitur est, ut in ipſam a c cadat. quod
demonſtrandum propoſuimus.

LEMMA III.

Sit parabole, cuius diameter a b: atque eam cŏtingen
tes rectæ lineæ a c, b d;
a c quidem in puncto c, b d ue
ro in b:
& per c ductis duabus lineis; quarum alter a c e
diametro æquidiſtet, alter a c f æquidiſtet ipſi b d:
ſuma
tur quod uis punctum g in diametro:
fiatque ut f b, ad
b g, ita b g ad b h:
& per g h ducantur g k l, h e m,
æquidiſtantes b d:
per m uero ducatur m n o ipſi a c
æquidistans, quæ diametrum ſecet in o:
& per n ducta
n p uſque ad diametrum, ipſi b d æquidistet.
Dico h o
ipſius g b duplam eſſe.
V_EL_ igitur linea m n o ſccat diametrum in g, uel in alĳs pun-
ctis:
& ſi quidem ſecat in g, unum at que idem punctum duabus li-
teris go notabitur.
Itaque quoniam f c, p n, h e m ſibiipſis æqui
distant:
& ipſi a c æquidiſtat m n o: fient triangula a f c, o p n,
o h m inter ſe ſimilia.
quare erit o h ad h m, ut a f ad fc: & per-
4. ſexti.mut ando o h ad a f, ut h m ad fc.