136 



CONIC SECTIONS. 



is similar to GCF, (29. and 15. 1. E.) is also similar to 

 FCL, (S. 6. E.) Therefore, LC : CF : : FP : PE or CH, 

 hence, LC.CHs=CF.FP, (16. 6. E.); but since the circle 

 must meet the line ED e, CH cannot exceed LC ; and 

 therefore CH 2 cannot exceed LC.CH, although it may- 

 be either equal to it or less. Hence, CH 2 may be 

 equal to any space that does not exceed CF.FP ; and 

 the point H may be on either side of F f. 



In the second case, (Fig 2.) from the position of the 

 points F, f, in respect of the lines PQ, p q, the point H 

 necessarily falls between C and G, and therefore is al- 

 ways within the circle ; hence it may be any where in 

 a line perpendicular to F fat C. 



Composition. In each case, draw a line to bisect 

 F f at right angles in C, and, in Case 1. (Fig. 1.) take 

 CB and C b in that line, each equal to a mean propor- 

 tional between CF and FP, and take H any where in 

 B b ; but, in Case 2. (Fig. 2.) H may be taken any- 

 where in the line which bisects Ff at right angles in 

 C. In each case, through FI draw a line parallel to ¥f, 

 meeting PQ and p q in E and e, and draw the lines 

 EFG, e fG, meeting in G, which will be a point in the 

 perpendicular to F^at C, (as shewn in the first proposi- 

 tion.) Describe, next, a circle about the triangle FGf, 

 and, in Case 1. let this circle meet CB in L, (Fig. 1.) 

 then if FL be drawn, it may be shewn, as in the Ana- 

 lysis, that LC.CH=CF.FP=:CB 2 (by construction) ; 

 therefore B falls between H and L, and so CL is not 

 less than CH ; but, in Case 2. (Fig. 2.) the point PI 

 falls always between C and G; therefore, in either case, 

 the circle must meet the line E e, Fig. 1. or that line 

 produced, (Fig. 2.) ; let D be either of the points of 

 section. Draw DF, Y)f } and the problem is con- 

 structed. 



Draw DG meeting Ff'm K ; and because the chords 

 FG, fG, and consequently the arcs FG,/G, are equal, 

 in Fig. 1. the angles GDF, GD/are equal, (27- 3. E.); 

 and, in Fig. 2. the angle GDF is equal to GF f, 

 (27. 3. E.) ; but in Fig. 2. KD /' is also equal to GF/; 

 (13. l.and22.3.E.); and therefore GDF=KD/: hence 

 in each case GK makes equal angles with DF, D/*,- 

 and therefore DF : D/: : KF : K/, (3. andA.6 E.) But 

 the lines E e, Ff being similarly divided at D and K, 

 (Geometry,) KF : K f : : DE : De, therefore DF : 

 Df: : DE : D e. 



Again, the angle GED being equal to GF/ (29. 1. E.) 

 «r GfF, that "is to GDF, (21. 3. E.) the angles FED, 

 KDF are equal, now the angles FDE, DFK are equal, 

 (29. I.E.); therefore the triangles EDF, DFK are 

 equiangular, (32. 1. E.) and 



ED : DF : : DF : FX, 



ED : FK : : ED 2 : DF 2 (2 Cor. 20.6.E.) 



: EG : GF : : PC : CF (4. & 2. 6. E.) 

 CA 2 : CF 2 (2 Cor. 20. 6. E.) 

 : : ED 2 : DF 2 

 ED:DF::eD:D/: 



Cor. 1 . This problem is indeterminate, or admits of 

 an indefinite number of solutions. For it appears that, 

 in Case 1. (Fig. 1.) the line EDe, in which the point 

 D is always situated, may have any position between 

 the two limits B, b ; and that in Case 2. (Fig. 2. ) it 

 may have any position whatever, provided, however, 

 that in each case it be parallel to F^ 



Cor. 2. For any one position of the line ED e, in 

 which it does not pass through B or b, in Case 1, there 

 we two points D, D', and no more : because a circle 

 can cut a straight line only in two points. When the 

 tine ED e passes through B or 6, in Case 1. there is only 

 one point. 



\ 



SCHO 



HUM. 



hence 



but ED : FK : 



and PC : CF : : 



therefore, CA 2 : CF 2 



and CA : CF : : 



As it appears from the construction of the problem, 

 that in both cases, there are innumerable points D, ha- K^ A J, K r 

 ving the property required, viz. two in any line meet- 

 ing B h between the points B, b, and parallel to F f, F 'g- !»*» 

 (Fig. 1.) or in any line whatever parallel to F/j (Fig. 2.) 

 these points must all he in some line XD Y of a deter- 

 minate kind, which will be what is called the locus of 

 the point that satisfies the conditions of the problem, 

 and the properties of which may be made the subject 

 of a geometrical theory. 



The figure of the locus depends greatly upon the 

 position of the points F, f in respect of the lines PQ, 

 p q, or (which comes to the same tiling,) the kind of 

 ratio that FD has to DE, andyT> to D c. For we have 

 seen in Fig. 1. where CF is less than CA, and conse- 

 quently the ratio of CF to CA, or of FD to DE is a 

 ratio of minority, that the locus extends only to a limited 

 distance on each side of the line F f: and that in Fig. 2. 

 where CF is greater than CA, and the ratio FD to DE 

 is a ratio of majority, the locus recedes to an indefinite 

 distance on each side of Ff. This peculiarity of figure 

 shews that there are two distinct species of lines, one 

 is the locus of D when FD has to DE a ratio of mino- 

 rity, and another, when it has a ratio of majority. 



Moreover, as in each case, we may suppose the ratio 

 to approach as near to that of equality as we please, 

 there must be an intermediate species, which may be 

 regarded as the limit to either of the other two. Now, 

 since (Fig. 1. and 2.) Ff: Pp ( : : CF : CP : : CF 2 : 

 CA 2 ) : : FD 2 : DE 2 ; when this last ratio approaches to 

 that of equality, the lines F^ P p must also approach to 

 equality ; so that if we suppose the point F and line 

 PQ to be given by position, the other point f and line 

 p q will recede from them continually ; and the ratio 

 of FD to DE becoming absolutely that of equality, the 

 point f and line p q will have gone oif to an infinite 

 distance, or, in other words, they will no longer exist. 



These three species of lines were denominated Co- 

 nic Sections by the ancient geometers, because they 

 may also be defined by the section of a cone and plane. 



They gave the name Ellipse to the line which is the 

 locus of D, when FD has to DE a ratio of minority; 

 they called it an Hyperbola, when the ratio was that 

 of majority ; and a Parabola, in the case of a ratio of 

 equality. We shall now explain the principal proper- 

 ties of these lines in the following Sections. 



SECTION II. 



Of the Ellipse. 



Definitions. 



1 . Let PEQ, p e q be two parallel straight lines given p LATg 

 by position, and F,f two given points between these CCVII. 

 lines in Pp a line perpendicular to them, equally dis- Fjcr 5 

 tant from C the middle of that line; and in Pp, let °" 

 CA, C a be taken in contrary directions, each equal to 



a mean proportional between CF and CP, then, if a 

 point D be supposed to move in the plane of the lines, 

 in such a manner, that DF, its distance from one of 

 the given points, has to DE, its distance from the given 

 line adjacent to that point, the given ratio of CF to 

 CA, and consequently (Sect, I. prop. 1 .) so that Df 

 its distance from the other point, has to D e, its dis- 

 tance from the other line, the same given ratio; the point 

 D will describe a curve line called an ellipse. 



2. Each of the linee PQ, p q is called a directrix^ 





