Full text: Briefwechsel zwischen Leibniz, Jacob Bernoulli, Johann Bernoulli und Nicolaus Bernoulli (1. Abtheilung, Band 3)

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38 
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ns. ni 
III. Theorema Cat-Optricum. Diametro (fig. 6) B M = 
j B F (quae radius est circuli curvam B C G in B osculantis) descri 
batur circulus AB CM, et radiet punctum A in puncta curvae cu- 
jusvis B C G in distantia B, C, per radios AB, AC; dico, si punctum 
A fuerit in peripheria circuli B CM, radios reflexos BI, CII fore 
parallelos: si extra circulum, convergentes: si intra, divergentes. 
Et reciproce, si radii incidentes contigui IB, IIC sint paralleli, coi 
bunt ipsorum reilexi BA, CA in puncto aliquo circuli BCM etc. 
Demonstrat. Productae sint particulae curvae in tangentes 
DBCL, ECG, eritque LBI = DBA = BAC 4 B C A = B M C 
(2BFC)4BCA = 2 ECD 4BCA = ECD4 ECA = LCG + GCH 
= LCH. Ergo BI parallela CII. Quod si autem sit intra circu 
lum, erit DBa — D B A = L BI, quare divaricabitur a C H. Sin a 
sit extra circulum, erit DBa— 1 DBA = LBI, quare coibit cum 
CH. Q. E. D. 
Cor oli. Hinc possunt inveniri puncta Causticae: Nam quia 
B F = 2 B M, et ang. B A M rectus, bine ex F centro circuli os- 
culatoris tantum perpendicularis FI vel F P demittenda in radium 
incidentem BI, vel reflexum BP, determinabitque dimidia BI vel 
BP punctum A in Caustica: puta si radii incidentes BI, CH fuerint 
paralleli. 
Quod si punctum A (fig. 7) radiet ex finita distantia, et ra 
diorum reflexi convergant, erit B A C + BIIC = 2 B F C. Demonstr. 
BAC4-BHC = DBA (LBH) — DCA -f- BHC = LBII — ECA 
4- ECD + BHC = LBH — GCH 4- LCG 4 BH C = LCII — BHC 
— GCH + LCG 4- BHC = LCH— GCH 4- LCG = LCG4-LCG 
— 2LCG = 2BFC. Q. e. d. Hinc inveniri potest relatio puncti 
H ad punctum F ita: Quia BAC —BMC, et BHC = BPC, erit 
BMC-f-BPC = 2BFC; sed BMC. B P C : : C P. CM (in infinite 
C p v r p r 
' A etBFC. BPC :: CP. CF, hoc 
parvis) hoc est, BMC 
CM 
C P v R P r / 
est, BFC - -g," - , quare BMC 4 BPC / = 
+ 
CM x BPC 
CF 
2CPX BPC 
C P x B P C 
CM 
CM 
CF 
, 4 CP4-CM 2CP , 
hoc hoc 
CMx CF 
- et quia CP . CH :: CM . CA, erit CH = 
est CP _ 
2CM —CF 
C A X CF 
2CM —CF' Constr * Ex P un cto radiante A ducatur ad CF
	        
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