Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

43. Lect . XII.

IN ſuſcepto negotio progredimur; quod ut (quatenus licet) decurte-
mus, verbíſque parcamus; obſervetur, in ſequentibus ubique _line-_
_am_ AB _curvam_ eſſe (quales tractamus) quampiam; cujus _Axis_ AD; huic applicatas omnes rectas BD, CA, MF, NG perpendiculares; & ME, NS, CB parallelas eſſe; _punctum_ M liberè ſumi; _arcum_
MN indefinitè parvum eſſe; rectam α β curvæ VB, α μ curvæ AM,
μ ν _arcui_ MN æquales eſſe; ad rectam α β applicatas ei perpendicu-
lares eſſe. His præſtratis,

43.1.

_Praparati@_
_Communis_.

I. Sit MP curvæ AB perpendicularis; & lineæ KZ L, α φ δta-
les, ut FZ ipſi MP, & μ φ ipſi M Fæquentur; erît _ſpatium_ α β δ ipſi
AD LK æquale.

43.1.

Fig. 156,
157.

Nam _Triangula_ MRN, PFM ſimilia ſunt, adeoque MN. NR
: : PM. MF. unde MN x MF = NR x PM, hoc eſt (ſubſtitutis
æqualibus) μ ν x μ φ = FG x FZ; ſeu rectang. μ θ = rectang. FH; ſpatium verò α β δ minimè differt ab indeſinitè multis rectangulis,
qualia μθ & ſpatium AD LK totidem rectangulis, qualia FH, æ-
quivalet. unde liquet Propoſitum.

II. Hinc, ſi curva AMB circa axem AD rotetur, habebit ſe _pro._ _ducta ſuperficies_ ad _ſpatium_ AD LK, ut _Circumferentia circuli Ad ra-_
_dium_; unde noto ſpatio AD LK cognoſcetur dicta _ſuperficies._ Con-
ſequentiæ rationem jam anteà pridem aſſignavimus.

43.1.

Fig. 156.

III. Exhinc _Spbæræ, Spbæroidis_ utriuſque, _Conidúmque ſuperficies_
_dimenſionem_ accipiunt; nam ſi AD ſit conicæ ſectionis, à qua iſtæ
figuræ oriuntur, axis; linea KZL ſemper aliqua conicarum exiſtet,
haud difficili negotio determinabilis. Hoc ſuggero tantùm, quoniam
nunc evulgatum habet ur.

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