. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. VIII. CAtherefore, CT : CF ::and by add. and sub. TFor/E, by Prop. VI. But, by sim. tri. TF : T/ :: FE : fe ;therefore, fE ~fc, and consequently , because FE is parallel to^-, the Ze = ZFET ;therefore, the Z FET = ZfEc. Q. E. D. CD :: CT r CA ;CA : CA —FE ;: T/ :: FE : 2CA—FE, Cor. As opticians find, that die angle of incidence is e-qual to the angle of reflection, it appears from our Proposi-tion, that rays
. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. VIII. CAtherefore, CT : CF ::and by add. and sub. TFor/E, by Prop. VI. But, by sim. tri. TF : T/ :: FE : fe ;therefore, fE ~fc, and consequently , because FE is parallel to^-, the Ze = ZFET ;therefore, the Z FET = ZfEc. Q. E. D. CD :: CT r CA ;CA : CA —FE ;: T/ :: FE : 2CA—FE, Cor. As opticians find, that die angle of incidence is e-qual to the angle of reflection, it appears from our Proposi-tion, that rays of light issuing from one focus, and meetingthe curve in every point, will be reflected into lines drawnfrom the other focus. So the ray fE is reflected into this is the reason why the points F,/, are called fon,or burning points^ CONIC SECTIONS. 295 PROPOSITION XI. If a line be drawn from either focus perpendicular toa tangent to any point of the curve, the distance of theirintersection from the centre will be equal to the semitrans-verse axis. That is, if FP, fp, be perpendicular to the tangent TP/>,then shall CP and C/>, be each equal to CA or For through the point of contact E draw FE, and/Emeeting FP produced in G. Then the ZGEP=ZFEP,being each equal to the Z/E/>, and the angles at P beingright, and the side PE being common, the two trianglesGEP, FEP, are equal in all respects, and so GE=FE, andGP=FP. Therefore, since FP=± FG, and FC=| F/,and the angle at F common, the side CP will be = | fQ orI AB, that is, CP=CA or CB. And in the same manner Cp—CA or CB. Q. E. D. Cor. i. A circle described on the transverse axis, as adiameter, will pass through the points P, p ; because allthe lines CA, CP, Cp, CB, being equal, will be the radii ofthe circle. Cor. 2. CP is parellel to/E, and C/» parallel to FE. 296 MATHEMATICS. Cor. 3. If at the intersections of any tangent, withthe circumscribed circle, perpendiculars to the tangent bedrawn, they will meet t
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