Huygens, Christiaan - Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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47338ΕΞΕΤΑΣΙΣ CYCLOM. tro Π P. Ducto deinde plano per A Π P, & alio huic pa-
rallelo D Φ S ſecundum lineam Φ S, erit jam pars ungulæ
hiſce duobus planis terminata, æqualis ſolido quod fit ex du-
ctu plani S Φ Π P in ſe ipſum; & pars ungulæ E D S Φ, æ-
qualis ei ſolido quod fit ex ductu plani E Φ S in ſe ipſum.
Quare nunc demonſtrandum erit duntaxat, partem E D S Φ
eſſe ad partem Φ A P ut 53 ad 203, Sit Φ N parallela E P,
& N C parallela Π A. Ergo quoniam ex proprietate Para-
boles, P N eſt {3/4} Π P, erit quoque P C {3/4} A P. Verùm &
S D æquatur {3/4} A P, quum ſit huic parallela , ſitque 1111. 16.
Elem. rabola E A F: Itaque junctâ C D, ea parallela & æqualis
erit lineis P S, N Φ. Ducatur ſecundùm D C planum
D B C parallelum baſi E Π F, fietque ſemiparabola B D C
æqualis & ſimilis ſemiparabolæ Π Φ N; & erit Φ B N di-
midius cylindrus parabolicus: D A C B verò dimidiata un-
gula. Hæc autem æquatur ſicut antea oſtendimus, duabus
quintis cylindri dimidiati, baſin habentis D B C & altitudi-
nem B A. Ergo quum ſemicylindrus Φ B N habeat altitu-
dinem B Π triplam ipſius B A, erit ungula dimid. D A C B
ad ſemicyl. Φ B N, ut 2 ad 15, hoc eſt, ut 8 ad 60.

Junctâ porro Φ Π, conſtat ſemiparabolam Π Φ N ad trian-
gulum Π Φ N eſſe ut 4 ad@ 3; ſed triangulus Π Φ N eſt ad
rectangulum Φ P ut 1 ad 6, (eſt enim baſis Π N tertia pars
ipſius N P) hoc eſt, ut 3 ad 18. Ergo ex æquo erit ſemi-
parab. Π Φ N ad rectang. Φ P ut 4 ad 18. Itaque & ſemi-
cylindrus Φ B N eſt ad parallelepipedum ejuſdem altitudinis
ſuper baſi Φ P, ut 4 ad 18. Dictiautem parallelepipedi di-
midium eſt priſma D N S; ergo ſemicylindrus Φ B N eſt ad
priſma D N S, ut 4 ad 9, hoc eſt, ut 60 ut 135. Qua-
lium igitur partium dimidiata ungula D A C B erat 8, ta-
lium ſemicylindrus parab. Φ B N erat 60, (ut ſuprà oſten-
ſum eſt) taliumque priſma D N S erit 135. Ac proinde ſo-
lidum A Π S D, quod ex iſtis tribus componitur, erit 203.
Eſt autem ungula dimidiata A D C B ad dimidiatam un-
gulam E A P Π, ut 1 ad 32, ſicut Cl. Vir. demonſtravit in
prop. 95. lib. 9. Ergo qualium partium ungula dimid.

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