CONIC SECTIONS. 



135 



Elements of Conic Sections, Sec. by Charles Hutton, 

 LL.D. London, 1787. 



Sectionum Conicarum Lilri Scptem. Accedit Tracta- 

 tes de Scctionibus Contois, et dc scriptoribus qui earum 

 doctrinam tradiderunt. Auctore Abramo Robertson, 

 A.M. Oxonii, 1792. 



A Geometrical Treatise on Conic Sections, in four 

 books, &c. by the Rev. Abram Robertson. Oxford, 

 1802. 



A Short Treatise on the Conic Sections, in which 

 the three Curves are derived from a general descrip- 

 tion on a Plane, by the Rev. T. Newton. Cambridge, 



1794. 



A System of Conic Sections adapted to the Study of 

 Natural Philosophy, by the Rev. D. M. Peacock. 1810. 



A Compendious and Practical Treatise on the Con- 

 struction, Properties, and Analogies of the Three Conic 

 Sections, by the Rev. D. Bridge. 1811. 



Essai de Geometrie Analylique applique aux Courhes 

 et aux Surfaces du second ordre, par J. B. Biot. Paris, 

 1810. 



A Treatise on Lines of the Second Order. This is 

 part of a work entitled, Geometry of Curve Lines, by 

 John Leslie, Professor of Mathematics in the Universi- 

 ty of Edinburgh. 



There is much valuable matter relating to the conic 

 Sections in several works, which do not treat expressly 

 on the subject. Particularly, in Newton's Principia, 

 lib. i. The leanied Jesuits, Le Seur and Jacquier, have 

 given a concise treatise in their commentary on the 

 work, at Prop. 8. lib. i. Maclaurin has treated of the 

 conic sections in his Geomctria Organica, sect. 1.; in his 

 Fluxions, chap. xiv. and in sect. 2. of the Appendix to 

 his Algebra. Euler has treated of them in his Intro- 

 ducfio in Analysin hifinilorum, lib. ii. cap. 5. ; and De 

 Moivre in his Miscellanea Analylica, lib. viii. cap. 2, 

 The Synopsis Palmariorum Matheseos of Jones, also 

 treats of the subject; but it would extend our catalogue 

 too much to name all the writers who have improved 

 the theory. The reader may see a copious list of them 

 in Bibliotheca Mathematica, Auctore Fred. Guil. Aug. 

 Murhard. Lipsae, 1798. 



The references in the following treatise are to be un- 

 derstood thus, (20. 1. E.) means the 20th Prop, of the 

 1st book of Euclid; (2. Cor. 20. 6. E.) means the 2d 

 Cor. of the 20th Prop, of the 6'th book ; again, (5.) 

 means the 5th prop, of the section in which the refe- 

 rence is found; (Cor. 1.) means the Cor. to the 1st 

 Prop. (2. Cor. 2.) means the 2d Cor. to Prop. 3. and 

 so on. 



SECTION I. 



'Preliminary Propositions. 

 Prop. 1. Theorem. 



Let PEQ, peq be two parallel straight lines given by 

 position, and F,f two given points in Fp, a line per- 

 pendicular to them, which are equally distant from C the 

 middle of that line, and let CA be a mean proportional 

 between CF and CP ; if D be such a point, that DF and 

 DE, its distances from one of the points and the adja- 

 cent line, have to each other the given ratio of CF to CA, 

 then also D/and D e, its distances from the other point 

 and line,. shall have to each other the same given ratio. 



Join EF, ef, producing them until they meet in G; 

 join GC, and draw GD, meeting F/in K, 



The triangles EPF, epfave manifestly equal (34. and 

 4. of 1. E.) therefore the angles PFE, pfe, and conse- 

 quently the angles GFf, G/F, GEe, GeE are equal 

 (15. and 29. of 1. E.) hence GF = G/ and GE = Ge 

 (6. 1. E.), and GC is perpendicular to F/(4. 1. E.) 

 Again the triangles EGD, FGK being similar, as also 

 the triangles FEP, FGC, we have 



ED : FK (: : EG : GF) : : CP : CF; 

 Now by hyp. DF : ED : : CF : CA, 



therefore ex. seq. inv. DF : FK : : CP : CA, (23. 5. E.'Y; 

 and since by hyp. ED : DF (: : CA : CF) : : CP : CA, 

 therefore, ED : DF : : DF : FK ; 



now the angles EDF, DFK are equal; (29. 1. E.) there- 

 fore the triangles EDF, DFK are equiangular (o. 6. E.) 

 and hence the angles FED, KDF are equal, and (in 

 both figures) the angle GED is equal to GDF : The 

 triangles GED, GDF are therefore equiangular, (32. 1. 

 E.) and GE : GD : : GD : GF, and hence also Ge ; 

 GD : : GD : Gf ; now the angle at G being common 

 to the triangles GcD, GT)f, they are equiangular (6. 

 6". E.) hence the angle GcD is equal to GD/ and (in 

 Fig. 2.) GeE is equal to ./' DK; but it has been shewn 

 that GEe is equal to FDG, therefore, in Fig. 1. the 

 angles FDK, ./DK are equal, and in Fig. 2. the angle 

 x DG is equal to yDK ; hence, and because the lines 

 F,e, Fyare similarly divided in DandK, in either case 

 FD :/D (: : FK :/K) : : DE : Be (3. and A. 6. E. and 

 ait. Geometry) and by alter. FD : DE : :fD : De. 



Corollary. The points F,^ D, G are in the cir- 

 cumference of a circle. For it has been shewn, that the 

 angle GDF is equal to GED, and therefore also to the 

 angle G/F ; hence (21. 3. E.) the points G, D, F,f, 

 must be in the circumference of a circle. 



Prop. II. Problem. 



As in last proposition, let PQ, p q be two parallel Piate 

 straight lines given by position, and F,f two given CCVIF. 

 points in P p, a line perpendicular to them, which are s " *' 

 equally distant from C the middle of that line, and let 

 CA be a mean proportional between CF and CP ; it is 

 required to find a point D such, that the ratio of its dis- 

 tances from each of the given points, and the line ad- 

 jacent to that point, shall have to one another the given 

 ratio of CF to CA. 



Analysis. Let us suppose the problem resolved, or 

 that D is found such, that DF and D f being joined, 

 and ED e drawn perpendicular to PQ and p q, then 

 DF : DE : : T>f: D e : : CF : CA. Draw FE and fe, 

 producing them until they meet in G. It follows from 

 the corollary to last Prop, that the points F, f, D, G, are 

 in the circumference of a circle ; and as this position 

 of the points appears to be all that is required, in order 

 to resolve the problem, we have only to give the line 

 E e such a position, that a circle described about the 

 triangle FG/' may meet it in a point D. Now, there 

 will be two cases of the problem. 



Case 1. When the points F,y*lie between the lines 

 PQ, pq, as in Fig. 1. 



. Case 2. When the lines PQ, p q, pass between the 

 points F, f, as in Fig. 2. 



In the first case, (Fig. 1.) join GC, producing it t» 

 meet the circle in L, and E e in H, and join FL ; then 

 by reasoning, as in Prop. 1. it will appear that the line 

 GCL is perpendicular to Ff, and therefore is the dia- 

 meter of the circle (I.3.E.) ; and consequently, GFL 

 is -a. right angle, (31.3.E.) and the triangle EPF, which 



