# Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

IV. Iiſdem ſtantibus, ſit curva AYI talis, ut ordinata FY ſit in-
ter congruas FM, FZ proportione media; erit _ſolidum_ ex ſpatio αδβ
circa axem α β rotato factum æquale _ſolido_, quod à _ſpatio_ ADI circa
axem AD converſo procreatur.

### 43.1.

Fig. 156,
157.

Nam eſt MN. NR: : PM. MF: : PM x MF. MF q: :FZ x
FM. MFq. unde MN x MFq = NR x FZ x FM; hoc eſt
μ ν x μ φ q = NR x FYq. Unde liquet Propoſitum.

V. Simili ratione colligetur, ſi FY ponatur inter FM, FZ _bime-_
_media_, fore _ſummam cuborum_ ex applicatis (quales μ φ) à curva α φ δ
ad rectam α β, æqualem _ſummæ cuborum_ ex explicatis à curva AYI ad
rectam AD. paríque modo ſe res habebit quoad cæteras _poteſta-_
_tes._

### 43.1.

Fig. 156,
157.

VI. Porrò, ſtantibus reliquis, ſit curva VXO talis, ut EX ipſi MP
æquetur; & curva πξψ talis, ut μ ξ æ quetur ipſi PF; erit ſpatium
α π ψ β æqua le ſpatio DV OB.

### 43.1.

Fig. 156.

Nam eſt MN. MR: : MP. PF; adeoque MN x PF = MR
x MP. hoc eſt μ ν x μ ξ = ES x EX. vel rectang. ET = rectang. μ σ. Unde liquet Propoſitum.

VII. Subnotetur hoc: Si curva AB ſit _Parabola_, cujus _Axis_ AD,
_parameter_ R; erit curva VXO _byperbola_, cujus _centrum_ D, _Axis_ DV,
cujuſque _parameter_ axi R æquatur (ſcilicet ob EXq = (PMq =
PFq + FMq = {R q/4}+FMq = {R q/4}+ DEq = ) DVq+ DEq). item _ſpatium_ α β ψ π erit _Rectangulum_; quoniam ſingulæ applicatæ
μ ξ ipſi {R/2} æquantur. Conſtat itaque dato _ſpatio byperbolico_ DVOB
curvam AMB dari; & viciſſim. Hoc obiter.

### 43.1.

Fig. 156.

VIII. Adnotari poſſet etiam omnia ſimul quadrata ex applicatis
ad rectam α β à curva π ξ ψ æquari rectangulis omnibus ex PE, EX
ad rectam DB applicatis (ſeu computatis); cubos ex μ ξ æquari ipſis
PFq x EX; ac ità porrò.

### 43.1.

Fig. 157.

IX. Adjungatur etiam (productâ PM Q) ſi ponatur FZ æqua-
lis ipſi PQ, & μ φ ipſi AQ; _ſpatium_ α β δ _ſpatio_ AD LK æ-
quari.

Fig. 157.

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