6666L*IBER* S*TATICÆ*
ad baſis terminos eductas biſecet F G, in G &
F, &
diametrum A D in H, &
ab ipſis biſectionum punctis ſint F I, G K parallelæ contra A D, quarum ver-
tices cum verticeſectionis & termino baſis proximo connectantur rectis I A,
I B, K A, K C; deinde eædem illæ F I, G K æquales (parallelæ diameter enim
A D, parallelarum I F, K G ſi ad B C baſin educantur ſeſquitertia eſſet per
19 propoſ. Archimed. de quad. parab. & ſublatis æqualibus, reliquæ I F, K G
æquales erunt) fecentur ratione dupla in L & M, tum recta L M connexa, in-
108[Figure 108]
terſecet diametrum A D in N, &
I K eandem in O.
Præterea tota diameter
A D ſecetur dupla ratione in P, parallela autem I F continuata occurrat baſi
B C in Q. Quandoquidem igitur A P dupla eſt ipſius P D, P erit trianguli
A B C gravitatis centrum, eadem ratione L, M, erunt centra gravitatis trian-
gulorum A B I, A C K, Ideoq́ue N (ſunt enim triangula æqualia) utriuſque
commune centrum, quare N P jugum erit, quod ſecetur in R, ut ratio N R
ad R P ſit eadem quæ trianguli A B C ad duo triangula A B I, A C K, hoc
eſt ut 4 ad 1 (parabola enim trianguli æquealti in eadem baſi ſeſquitertia eſt,
demonſtrante Archimede propoſ. 24. de quadratura paraboles. Simili planè viâ
fecetur parabola a b c.
ab ipſis biſectionum punctis ſint F I, G K parallelæ contra A D, quarum ver-
tices cum verticeſectionis & termino baſis proximo connectantur rectis I A,
I B, K A, K C; deinde eædem illæ F I, G K æquales (parallelæ diameter enim
A D, parallelarum I F, K G ſi ad B C baſin educantur ſeſquitertia eſſet per
19 propoſ. Archimed. de quad. parab. & ſublatis æqualibus, reliquæ I F, K G
æquales erunt) fecentur ratione dupla in L & M, tum recta L M connexa, in-
A D ſecetur dupla ratione in P, parallela autem I F continuata occurrat baſi
B C in Q. Quandoquidem igitur A P dupla eſt ipſius P D, P erit trianguli
A B C gravitatis centrum, eadem ratione L, M, erunt centra gravitatis trian-
gulorum A B I, A C K, Ideoq́ue N (ſunt enim triangula æqualia) utriuſque
commune centrum, quare N P jugum erit, quod ſecetur in R, ut ratio N R
ad R P ſit eadem quæ trianguli A B C ad duo triangula A B I, A C K, hoc
eſt ut 4 ad 1 (parabola enim trianguli æquealti in eadem baſi ſeſquitertia eſt,
demonſtrante Archimede propoſ. 24. de quadratura paraboles. Simili planè viâ
fecetur parabola a b c.
DEMONSTRATIO.
Vt A D ad A O, ſic per 20 prop.
1 lib.
Apoll.
quadratum D B ad quadra-
tum O I, hoc eſtad Q D, ſed Q D dimidia eſt ipſius B D, nam F Q parallela
contra A D biſecat inſcriptam A B, quadratum itaque D Q hoc eſt O I ſub-
quadruplum erit quadrati B D, & ſegmentum igitur A O {1/4} erit totius A D, cui
O H æqualis eſt, nam integra A D biſecatur in H, & N H {1/12} ejuſdem, quæ
ad H D {1/2} addita exhibet N D {7/12} de qua deducta P D {1/3} relinquet P N {1/4}, fed
R P ſubquadrupla eſt ipſius N R, & totius igitur A D ſubvigecupla, quæ ad-
dita ad P D {1/3} dabit D R {23/60} & reliquam R A {37/60}. Quamobrem ut 37 ad 23 ſic
A R ad R D. eodem modo evincetur ſegmenta alterius parabolæ a r, r d, eſſe
ut 37 ad 23. Itaque rectilinea ſimili ratione in diſſimilibus parabolis inſcripta
centrum gravitatis habent in diametris, à quibus ipſæ diametri in homologa
ſegmenta dividuntur. Ac denique ſi in parabolæ ſegmentis B I, I A, A K, K C
triangula itidem ut in ſegmentis B I A, A K C inſcribantur, & rectilineorum
gravitatis centra S & ſ inveniantur, tandem ſimiliter concludes A S, S R, ſeg-
mentis aſ, ſr proportionalia eſſe, verum infinita hujuſmodi inſcriptione con-
tinuô ad E & e propius acceditur. Itaque hujuſmodi rectilineorum γνωζί-
11Deſinit
Archimed.
prop. 1. lib. 2.
iſerrhopiewr. μως ſcitè (ut cum Archimeàe loquar) in parabolas inſcriptorum gravitatis cen-
tra, diametros A D & a d in ſegmenta homologa perpetuò tribuent; atque
adeò ipſæ quibus inſcribuntur parabolæ A B C, a b c, ſegmenta diametri
tum O I, hoc eſtad Q D, ſed Q D dimidia eſt ipſius B D, nam F Q parallela
contra A D biſecat inſcriptam A B, quadratum itaque D Q hoc eſt O I ſub-
quadruplum erit quadrati B D, & ſegmentum igitur A O {1/4} erit totius A D, cui
O H æqualis eſt, nam integra A D biſecatur in H, & N H {1/12} ejuſdem, quæ
ad H D {1/2} addita exhibet N D {7/12} de qua deducta P D {1/3} relinquet P N {1/4}, fed
R P ſubquadrupla eſt ipſius N R, & totius igitur A D ſubvigecupla, quæ ad-
dita ad P D {1/3} dabit D R {23/60} & reliquam R A {37/60}. Quamobrem ut 37 ad 23 ſic
A R ad R D. eodem modo evincetur ſegmenta alterius parabolæ a r, r d, eſſe
ut 37 ad 23. Itaque rectilinea ſimili ratione in diſſimilibus parabolis inſcripta
centrum gravitatis habent in diametris, à quibus ipſæ diametri in homologa
ſegmenta dividuntur. Ac denique ſi in parabolæ ſegmentis B I, I A, A K, K C
triangula itidem ut in ſegmentis B I A, A K C inſcribantur, & rectilineorum
gravitatis centra S & ſ inveniantur, tandem ſimiliter concludes A S, S R, ſeg-
mentis aſ, ſr proportionalia eſſe, verum infinita hujuſmodi inſcriptione con-
tinuô ad E & e propius acceditur. Itaque hujuſmodi rectilineorum γνωζί-
11Deſinit
Archimed.
prop. 1. lib. 2.
iſerrhopiewr. μως ſcitè (ut cum Archimeàe loquar) in parabolas inſcriptorum gravitatis cen-
tra, diametros A D & a d in ſegmenta homologa perpetuò tribuent; atque
adeò ipſæ quibus inſcribuntur parabolæ A B C, a b c, ſegmenta diametri
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