Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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[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.
[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.
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168FED. COMMANDINI ſunt uertice, eandem proportionem habent, quam ipſarũ
baſes.
eadem ratione pyramis a c l k pyramidi b c l k: & py
ramis a d l k ipſi b d l k pyramidi æqualis erit.
Itaque ſi a py
ramide a c l d auferantur pyramides a clk, a d l k:
& à pyra
mide b c l d auferãtur pyramides b c l k, d b l K:
quæ relin-
quuntur erunt æqualia.
æqualis igitur eſt pyramis a c d k
pyramidi b c d _K_.
Rurſus ſi per lineas a d, d e ducatur pla-
num quod pyramidem ſecet:
ſitq; eius & baſis communis
ſectio a e m:
ſimiliter oſtendetur pyramis a b d K æqualis
pyramidi a c d K.
ducto denique alio piano per lineas c a,
a f:
ut eius, & trianguli c d b communis ſectio ſit c fn, py-
ramis a b c k pyramidi a c d K æqualis demonſtrabitur.

ergo tres pyramides b c d _k_, a b d k, a b c k uni, &
eidem py
ramidia c d k ſint æquales, omnes inter ſe ſe æquales erũt.
Sed ut pyramis a b c d ad pyramidem a b c k, ita d e axis ad
axem k e, ex uigeſima propoſitione huius:
ſunt enim hæ
axes cum baſibus æquales con
tinent angulos, quòd in eadem recta linea conſtituantur.

quare diuidendo, ut tres pyramides a c d k, b c d _K_, a b d _K_
ad pyramidem a b c _K_, ita d _k_ ad _K_ e.
conſtat igitur lineam
d K ipſius _K_ e triplam eſſe.
ſed & a k tripla eſt K f: itemque
b K ipſius _K_ g:
& c K ipſius K l tripla. quod eodem modo
demonſtrabimus.
Sit pyramis, cuius baſis quadrilaterum a b c d; axis e f:
& diuidatur e fin g, ita ut e g ipſius g f ſit tripla. Dico cen-
trum grauitatis pyramidis eſſe punctum g.
ducatur enim
linea b d diuidens baſim in duo triangula a b d, b c d:
ex
quibus intelligãtur cõſtitui duæ pyramides a b d e, b c d e:

ſitque pyramidis a b d e axis e h;
& pyramidis b c d e axis
e K:
& iungatur h _K_, quæ per ftranſibit: eſt enim in ipſa h K
centrum grauitatis magnitudinis compoſitæ ex triangulis
a b d, b c d, hoc eſt ipſius quadrilateri.
Itaque centrum gra
uitatis pyramidis a b d e ſit punctum l:
& pyramidis b c d e
ſit m.
ductaigitur l m ipſi h m lineæ æquidiſtabit: nam el ad
112. ſexti.