154 



CONIC SECTIONS. 



Hyper- 

 bola. 



F!e. 6?. 



Eg. ei. 



(2. Ccr. 19.), therefore NIT: LE : : HR: EQ, and by 

 alternation, NH : HR : : LE : EQ. Now the angles 

 at H and E are equal, therefore the triangles NHR, 

 LEQ are equiangular, and NR is parallel to LQ • 

 consequently, RP is an ordinate to the diameter CQ 

 (13.), and is bisected by it at G ; and as CQ bi- 

 sects all lines which are parallel to KL, and are 

 terminated by the hyperbola, it will bisect the area 

 PQR. Let the equal areas PQG, RQG be taken from 

 the equal triangles PCG, RCG, and there will re- 

 main the hyperbolic sectors PCQ, RCQ equal to each 

 other. Therefore (33.) the areas DPQE, EQRH are 

 also equal. 



Cor. Hence, if CD, CE, CH, &c. any number of seg- 

 ments of the asymptote be taken in continued propor- 

 tion, the areas DPQE, DPQRH, &c. reckoned from 

 the first line DP, will be in arithmetical progi-ession. 



Prop." XXXV. Problem. 



An hyperbola being given by position, to find its 

 axes. 



Let HA/* be the given hyperbola. Draw two paral- 

 lel straight lines Hk, Kit, terminating in either of the 

 opposite branches, and bisect them at L and M ; join 

 LM, and produce it to meet the hyperbola in P ; then 

 LP will be a transverse diameter (4. Cor. IS.) Let p 

 be the point in which it meets the opposite branch, bi- 

 sect Vp in C ; the point C is the centre (9.) Take D, 

 any point in the hyperbola, and on C as a centre, with 

 the distance CD, describe a circle ; if this circle be 

 wholly without the opposite branches of the hyperbo- 

 la, then CD must be half the transverse axis (25.) ; but 

 if not, let the circle meet the hyperbola again in d, 

 join Del, and bisect it in E, join CE meeting the oppo- 

 site branches in A and a ; then An will be the trans- 

 verse axis (5. Cor. 13.), for it is perpendicular to Dd 

 (3. 3. E.), which is an ordinate to Act. The other axis 

 will be found, by drawing Bb a straight line through 

 the centre perpendicular to Aa, and taking CB so, that 

 CB a may be a fourth proportional to the rectangle 

 AE.E<7, and the squares of DE and CA, thus CB is half 

 the conjugate axis, (24.) 



SECTION IV. 

 Of the Parabola. 



Definitions. 



1. Let PQ be a straight line given by position, and F 

 a given point without it, and let a point D move in their 

 plane, in such a manner that DF, its distance from the 

 given point, is equal to DE, its distance from the given 

 line ; the point D will describe a curve line, called a 

 Parabola. 



2. The straight line PQ is called the Directrix. 



3. The given point F is called the Focus. 

 Corollaries to Def. 1, 2, and 3. If a straight line FP 



be drawn through the focus perpendicular to the direc- 

 trix at P, the parabola passes through A, the middle of 

 FP, and it cannot meet the perpendicular FP in any 

 other point. 



2. If MF m be drawn perpendicular to FP, and FM 

 and F m be each taken equal to FP, the parabola will 

 pass through M and m, and through no other points 

 *n the line M m, 



Def. 4. The straight line AF produced indefinitely, Parabola, 



is called the Axis. ~— ~y-"«— ' 



5. The point A, in which the axis meets the curve, 

 is called the Vertex. 



Scholium. 



From the definition of the curve, we have the fol- 

 lowing method of describing it mechanically. Place 

 the eclge of a fixed ruler along PQ the directrix, and 

 to this edge apply another moveable ruler, LEG, of 

 any length which is so constructed that one of its sides 

 EG is always perpendicular to EQ. Fasten one end 

 of a string equal in length to GE at G the end of this 

 ruler, and the other at F the focus ; and pull the string 

 tight by passing it round a pin D, which slides along 

 GE. If the moveable ruler be now made to slide along 

 the fixed ruler, and the pin D along the line EG, so as to 

 keep the string always extended into the two straight 

 lines GD, DF, it is evident that the pin will trace the 

 parabola. 



Prop. I. 



If a circle be described on F the focus as a centre, 

 with a radius equal to FP its distance from the direc- 

 trix ; and from any point D in the parabola a straight 

 line be drawn to the focus and produced to meet the 

 circle in I, and IN be drawn perpendicular to the axis; 

 the rectangle FD.PN is equal to the constant space 

 PF\ 



Draw DE perpendicular to the directrix, and join 

 EF, PI. The triangles EDF, PFI are similar (6. 6. E.) 

 for ED=DF, (Def. 1.) and PF=FI, and the angles 

 EDF, PFI are equal (29- 1; E.) therefore FE is paral- 

 lel to PI (28.1.) The triangles FPE, PNI are also 

 equiangular; for the angles at P and N are right angles, 

 and the angles EFP, NPI are equal (29. 1. E.) Hence 

 DF : FE : : IF : IP : and FE : FP : : IP : PN (4. 6. E.) 

 therefore, ex req. DF : FP : : IF : PN (22. 5. E.) and 

 DF.PN=FP.F1~FP ! (16.6. E.) 



Cor. Hence the point in which a line drawn from 

 the focus meets the curve may be found, viz. by tak- 

 ing FD a third proportional to PN and FP. 



i Scholium. 

 From this Proposition we learn what is the figure of 

 the curve, by considering the change that takes place 

 in the magnitude of the line FD, corresponding to a 

 change in its position. When FD has the position FA, 

 the points I and N are at L, and as in this case PN, 

 one side of the constant rectangle FD.PN is the great- 

 est possible ; the other side FD must be the least. 

 Suppose now FD to depart from the position FA, and 

 to revolve about F; then, as the angle PFD, or IFL 

 increases, the point N will approach to P, and as PN 

 one side of the rectangle FD.PN now decreases until 

 it vanishes, FD its other side must increase ; and go 

 on increasing until it has attained a magnitude greater 

 than any that can be assigned. In general, we may 

 conclude, 



1. That the parabola is a continuous curve, which 

 passes through A the middle of FP, and extends to an 

 indefinite distance from the focus on both sides of the 

 axis. 



2. That every line which can be drawn from the fo- 

 cus will, if produced, terminate in the curve, with 

 the exception of FX the pi-olongation of FA ; and that 

 the least will be FA, and of the others, those nearer 

 the least will be less than those more remote. 



Fi«r. Cl. 



Fi.tr- ea 



