Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

LFK ſeſe contingunt. quare curvæ DIF, KFK ſe quoque con- tingent. ergò denique recta FS curvam DIF continget.

35.1.

22. Lect.
VII.
Hyp.
4. Lect.
VIII.
2. Lect.
VIII.

VIII. Quòd ſi rectæ DF quamvis aliam conſtanter eandem ad ar-
cus AE rationem obtinuerint, itidem deſignari poteſt recta curvam
DIF tangens, ex hac, & ſeptima octavæ Lectionis; erit utique tan-
gens iſta huic FS parallela.

IX. Hinc nedum _ſpiralis circularis_, aſt innumerabilium ſimili ratione
progenitarum aliarum curvarum _Tangentes_ determinantur.

X. Sint curva quæpiam AEH, recta AD (in qua determinatum
punctum D) recta DH poſitione data; ſit item curva AGB talis,
ut in hac aſſumpto quocunque puncto G, & per hoc ac D projectâ
rectâ DGE (quæ curvam AEH ſecet in E) ductâque GF ad DH
parallelâ habeant AE, AF aſſignatam rationem X ad Y; tangat au-
tem recta ET curvam AEH; recta deſignetur oportet, quæ curvam
AGB ad G tangat.

35.1.

Fig. 108

Fiat recta EV æqualis arcui EA; & concipiatur curva OGO ta-
lis, ut projectâ quâcunque rectâ DOL (quæ curvam OGO ſecet
puncto O, rectam ET in L) ductâque OQ ad GF parallelâ, ſit
VL. AQ: : X. Y; eſtque curva OGO (è ſuprà monſtratis) _Hy-_
_perboln;_ hanc tangat recta GS; etiam recta GS curvam AGB
continget.

Nam concipiatur altera curva NGN talis, ut cùm hanc ſecet recta
arbitraria DL in N, curvam AEH in K, rectam TE in L; ductáq; ſit NR ad GF parallela, ſit VL + LK. AR: : X. Y; manife-
ſtum eſt curvam NGN utramque curvam AGB, & OGO tange-
re. [ſecet enim recta DL curvam AEB in I, ducatúrque IP ad
GF parallela; quum ergò ſit VL + LK. AR: : X. Y: : AK. AP, & ſit VL + LK & gt; AK; erit AR & gt; AP; vel DR & lt; DP; adeóque DN & lt; DI; unde punctum N intra curvam AGB
ſemper cadet; ac proinde curva NGN curvam AGB tan-
get; ſimilique planè diſcurſu curva NGN curvam OGO contin-
get. ] Itaque curvæ AGB, OGO ſeſe (æquipollentèr) tangunt. Quare cùm recta GS curvam OGO tangat; eadem curvam AGB
quoque continget: Q. E. F.

Si curva AEH ſit circuli quadrans, cujus centrum D; erit curva
AGB _Quadratrix communis_. Ejus igitur _Tangens_ (unà cùm omni-
um ſimili ratione genitarum tangentibus) hoc pacto deſignatur,

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