Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

@ibus μνκμψ par) æquatur ſubduplo ſpatii PLOQ.

VII. Omnia quadrata ex rectis μψ (ad rectam αμ applicais) æquant
CA x CP x PX(hoc eſt _parallelipipedum Baſe Rectangulo_ ACPD,
_Altitudine_ CS).

45.1.

Fig. 167.

Hujus _Effati demonſtrationem_ (quanquam π& χΗ& ν) tranſilio; quo-
niam aliud _Scbema_ diſcursúmque præ reliquis pleríſque longiuſculum
expoſcit; neque rem tanti video.

VIII. Curva AYY talis ſit, ut FY æquetur ipſi AS; ductâ tum rectâ YI
ad AC parallela, erit etiam _ſpatium_ AC IY YA (hoc eſt _ſumma_
_Tangentium_ ad _arcum_ AM pertinentium, & ad rectam AC applica-
tarum, unà cum _rectangulo_ FCIY) æquale _ſubduplo ſpatio byperbo-_
_lico_ PL OQ.

45.1.

Fig. 166.

Nam _ſpatium_ α γ π μ æquatur _rectangule_ ACPD; hoc eſt _rectangulo_ FC IY (nam eſt CA. AS: : CF. FM; vel CAFY: :
CF. CP. adeoq; CA x CP = FY x CF). item ſpatium γπψ (hoc eſt omnes
rectæ TF ad αε applicatæ, quotquot ad arcum AM pertinent) æ- quatur _ſpatio_ AFY; ergo _ſpatium_ ACIYA æquatur _ſpatio_ αγψμ; hoc eſt (ut mox oſtenſum) _ſemiſſi ſpatii byperbolici_ PL OQ.

45.1.

1. Lect.
XII.
14. Lect.
XII.

Aliter illud, (eíque connexa) dimenſus ſum, _boc præmiſſo Lem-_
_mate._

IX. Sit _Hyperbola aquilatera_ (axes nempe pares habens) ERK ad
cujus axes CE D, CI; & ad hos ordinatæ KI, KD; ſit item curvâ
EVY talis, ut in _byperbola_ liberè ſumpto puncto R, ductâque recta
RVS ad DC parallelâ, ſint SR, CE, SV continuè proportiona-
les; connexâ rectâ CK, erit _Spatium_ CE YI _Sectoris byperbolici_
KCE duplum.

45.1.

Fig. 168.

Nam ducatur RT _byperbolam_ tangens, & R Had CI parallela. Eſtque CH. CE: : CE. CT. quare CT = SV; vel HT = RV. itaque _Spatium_ ED KY duplum eſt _ſegmenti_ EDK. item _rectangu-_
_lum_ IKDC _trianguli_ CDK duplum eſt; ergo _reliquum ſpatium_
CE YI _reliqui ſectoris_ ECK duplum eſt.

45.1.

10 Lect. XI.

X. Reſumptâ jam quadrante circulari AC B, ſit CE = CA; & axe AE, _parametro etiam_ AE, deſcripta ſit _Hyperbola_ EKK; poſitóque curvam AYY talem eſſe, ut ordinatâ quâcunque rectâ
MFY, ſit FY tangenti AS æqualis; ducatur recta YIK (rectam

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