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THE work now in part published is the result of an attempt to reduce the chaos of Geometrical Conics to order. The subject having suffered not a little from desultory treatment, I have endeavoured to reconstruct it on a uniform plan, taking as a standard whereby to regulate the sequence of proofs the principle that Chord-properties should take precedence of Tangent-properties, the latter being deduced from the former rather than the former from the latter.
This principle commends itself from more than one point. of view. In proofs of chord-properties it is superfluous to anticipate the comparatively complex notion of a tangent; while on the other hand the idea of a limit is the more clearly apprehended when approached in its natural order, and the properties of tangents are proved most convincingly when their intrinsic relation to the properties of chords. is shewn. But, above all, this order refers the student consistently to first principles, thus rendering Geometry at once more thorough as a means of intellectual training, and more effective as an introduction to the powerful modern methods of Analysis.
Simplicity is of course essential in an elementary work; but I have attempted at the same time to secure comprehensiveness by assigning their due importance to properties by which the Conic is characterized in the Higher Geometry of Curves-such as the quadratic relation between its coordinates and the rectilinearity of its diameters. As regards the practical working of this arrangement, it is scarcely too much to say that in the Parabola the student who has thoroughly grasped Prop. II. has little more to learn. In the Rectanb
gular Hyperbola the corresponding theorem leads by easy stages to the solution of all difficulties; while in the Ellipse and the general Hyperbola a single construction* suffices for the demonstration of all the leading properties of conjugate diameters.
Orthogonal Projection, already introduced under a new namet, will be further discussed in the sequel. The Right Cone will be treated somewhat less inadequately than heretofore; but, despite the skilful advocacy of Mr Stuart Jackson, I am unable to acquiesce in the primary definition of Conics from the solid. No allusion has yet been made to the Conjugate Hyperbola, which may be viewed as a contrivance for giving a false definiteness to the student's conceptions, and perpetuating his illusion that the Hyperbola is a discontinuous
For the proof of the chord-property that PN2 varies as AN. NA' my best thanks are due to Mr W. Allen Whitworth. The same proof was discovered independently by Mr Besant. In respect of the general tangent-property of Art. 108, I have pleasure in repeating my acknowledgments to Professor Adams. Mr Drew kindly places at my disposal his well-known proof that the tangent makes equal angles with the focal radii to its point of contact. In the Rectangular Hyperbola I have endeavoured to do justice to the investigations of Mr Wolstenholme, who has thrown much light upon that remarkable curve. I have profited by the advice and assistance of Mr Rawdon Levett of King Edward's School Birmingham, Secretary of the Association for the Improvement of Geometrical Teaching. It is needless to add that I am a debtor to Dr Salmon's inexhaustible works.
*The triangles pCN, dCR in Arts. 72, 212 differ only in relative position. + See page 38. Arts. 53, 202.
THE proofs in this work are so far independent that various propositions may be conveniently omitted in the first reading.
The order of the Chapters may also be varied. The Rectangular Hyperbola may with advantage be studied very early. Of all Conics it is that which in its properties is most nearly identical with the circle; and the conception of its form is simplified by the use of the asymptotes as guiding lines.
The figures for the general Hyperbola are so arranged that they may be compared with the figures for the Ellipse.