Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

Nam ob MN. NR: : PM. MF: : PQ. QA; erit MN x
QA = NR x QA; hoc eſt rectang. μ θ = rectang. FH.

X. Porrò, curvam AB tangat recta MT, ſintque curvæ DXO,
α φ δ tales, ut EX æquetur ipſi MT, & μ φ ipſi MF; erit ſpatium
α β δ æquale _ſpatio_ DXOB.

Fig. 158.
159.

Nam MN. MR: : MT. MF. quare MN x MF = MR x MT; hoc eſt μ ν x μφ = ES x EX; unde patet.

XI. Hinc rurſus, _ſuperficies ſolidi ex ſpatii_ ABD circa axem AD
converſione progeniti ad _ſpatium_ DX OB ſe habet, ut _Circuli Cir-_
_cumf._ ad _radium_; hoc igitur noto ſimul illa innoteſcet. unde rurſus
_Spbaroidum, Conoidumque ſuperficies_ dimetiri licebit.

43.1.

Fig. 158.

XII. Si linea DYI talis fuerit, ut ſit EY = √ EX x MF; erit
_ſolidum_ ex _ſpatio_ αβδ circa axem αβ rotato factum æ quale _ſolido, quod_
_ex ſpatio_ DBI circa axem DB rotato progignitur.

Etenim eſt MN. MR: : MT x MF. MF q: : EX x MF. MFq
: : EYq. MFq. quare MN x MFq = MR x EYq. hoc eſt μ ν
x μ φ q = ES x EYq.

43.1.

Fig. 158.
159.

XIII. Simili ratione _Cuborum (aliarumque poteſtatum)_ ex ordina-
tis μ φ _ſummas_ cum _ſpatiis_ ad rectam DB computatis licebit conferre.

XIV. Sint prætereà lineæ AZK, αξψ ætales, ut FZ ipſi MT, & μξ ipſi TF æquentur; _ſpatium_ αβψ æquabitur _ſpatio_ ADK.

Etenim MN. NR: : MT. TF; hoc eſt μ ν. FG: : FZ. μ ξ. quare μ ν x μ ξ = FG x FZ.

43.1.

Fig. 158.
159.

XV. Etiam _ſumma quadratorum_ ex qpplicatis μ ξ æquatur _ſummæ_
_Rectangulorum_ ex TF, FZ; & _ſumma Cuborum_ ex μ ξ æquantur
ipſis TFq x FZ (ad rectam ſcilicet AD computationem exigendo)
paríque quoad cæteras poteſtates modò.

43.1.

Fig. 158,
159.

XVI. Rurſus ponatur recta QMP curvæ AMB perpendicularis; ſitque recta β δ æqualis ipſi BD, & compleatur _Rectangulum_ αβδζ; tum curva KZL talis ſit, ut FZ ipſi QP æquetur; erit _rectang._ αβδζ
æquale _ſpatio_ AD LK.

43.1.

Fig. 160,
161.

Nam eſt MN. NR: : (PM. MF: :) PQIF. quare MN
x IF = NR x PQ; hoc eſt μν x μξ = FG x FZ. unde patet.

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