Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

51. Prop. 3.

Datus ſit _Conus rectus_ ABC _p._ Secetur à plano (puta _triangulo_
_qrt_) quod quidem planum ſecabit _axem coni_ in puncto _q_ ſupra _verti-_
_cem_ productum & in communi interſectione cum _ſuperficie coni_ habe-
bit _lineam byperbolicam_ RS_t_ ducantur à vertice coni A rectæ A _r_, A _t_,
à puncto _q_ demittatur perpendiculum _q_ X lateri coni A _p_ producto & à
puncto A perpendiculum AZplano _qrt._

51.1.

Fig. 178.

Dico _ſuperficies contca_ terminata à _linca byperbolica, rst_ & rectis
_r_ A, _t_ A, ita ſe habet ad _figuram byperbolicam cavam qrstq_ ut _perpen-_
_diculum_ AZad _perpendiculum q_ X.

Recta enim _qr_, circumlata, quieſcente termino _q_ per lineas _rst, t_ A, Ar
generat tres _ſuperficies_, nempe _byperbolicam cavam qr, st_, & _duo tri-_
_angula qt_ A, _q_ A _r_, quæ unà cum _ſuperficie conica_ terminata à lineis
_rst, t_ A, A _r_, comprehendunt _Solidum qrs, t_ A _r._ Hoc verò _ſolidum_
_œguale_ eſt _pyramidi_ cujus _altitudo_ eſt æqualis perpendiculo _q_ X, nam
infinitæ pyramides _q_ A _r_ V, _q_ AVV, exhauriunt ſolidum _qr_ S _t_ A _r._ Si verò aliter contemplari volumus, hoc ſolidum _qrst_ A _r_ poteſt con-
ſideraritanquam _ſigura @onica_ A _r_ S _tqr_ habens pro _baſe figuram by-_
_perbolicam_ cavam _qr_ S _tq_, & pro altitudine _perpendiculum_ AZ. Ergò
reciprocando _baſes altitudinibus_, ut AZad q X, ita _ſuperficies, r_ S t A _r_
ad _figuram byperbolicam cavam qr_ S _tq._

52. Prop. 4.

Datus ſit _Conus rectus_ AB _b g_ ſecetur à plano HFEGper axem
infra verticem, a puncto H ubi _planum_ fecat _axem coni_, demittatur HK
_perpendiculum_ lateri cuilibet coni & à verticè A _perpendiculum_ ALpla-
no HFE G.

52.1.

Fig. 179.

Dico, _Superſicies conica_ terminata a lineis FECGAAF ita ſe
habebit ad _planum_ HFEG ut _perpendiculum_ AL ad _perpendiculum_
H K.

Probatur eodem fere eodem fere argumento quo ſuperior.

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