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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14416FED. COMMANDINI
SIT pyramis, cuius baſis triangulum a b c; axis d e: &
ſecetur plano baſi æquidiſtante;
quod ſectionẽ faciat f g h;
occurratq; axi in puncto k. Dico f g h triangulum eſſe, ipſi
a b c ſimile;
cuius grauitatis centrum eſt K. Quoniã enim
1116. unde
cimi
duo plana æquidiſtantia a b c, f g h ſecantur à plano a b d;
communes eorum ſectiones a b, f g æquidiſtantes erunt: &
eadem ratione æquidiſtantes ipſæ b c, g h:
& c a, h f. Quòd
cum duæ lineæ f g, g h, duabus a b, b c æquidiſtent, nec
ſintin eodem plano;
angulus ad g æqualis eſt angulo ad
2210. undeci
mi.
b:
& ſimiliter angulus ad h angulo ad c: angulusq; ad f ei,
qui ad a eſt æqualis.
triangulum igitur f g h ſimile eſt tri-
angulo a b c.
At uero punctum k centrum eſſe grauita-
tis trianguli f g h hoc modo oſtendemus.
Ducantur pla-
na per axem, &
per lineas d a, d b, d c: erunt communes ſe-
3316. unde-
cimi
ctiones f K, a e æquidiſtantes:
pariterq; k g, e b; & k h, e c:
quare angulus k f h angulo e a c; & angulus k f g ipſi e a b
4410. unde-
cimi
eſt æqualis.
Eadem ratione
98[Figure 98] anguli ad g angulis ad b:
&
anguli ad h iis, qui ad c æ-
quales erunt.
ergo puncta
e _K_ in triangulis a b c, f g h
ſimiliter ſunt poſita, per ſe-
xtam poſitionem Archime-
dis in libro de centro graui-
tatis planorum.
Sed cum e
ſit centrum grauitatis trian
guli a b c, erit ex undecíma
propoſitione eiuſdem libri,
&
K trianguli f g h grauita
tis centrum.
id quod demonſtrare oportebat. Non aliter
in ceteris pyramidibus, quod propoſitum eſt demonſtra-
bitur.

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