It is also shown that Props. IX., X. are geometrically equivalent to the ordinary polar equation of a Conic; whilst Props. III., IV. lead to those of the tangent and chord respectively. The first of these results was pointed out by Professor Adams, to whom I am indebted for notes that have formed the basis of several proofs in the Chapter now under consideration. The above propositions are also useful in establishing theorems not usually proved by elementary geometrical processes (see Ex. 25, p. 22), whilst the interpretation of results is a manifest advantage to the student on his first introduction to analytical methods. The proposition QV2=4SP.PV, in the parabola, has been proved by assuming that PV=PT, and that the external angle between any two tangents is equal to that which either of them subtends at the focus; the latter being perhaps one of the most obvious deductions from the fundamental properties of tangents. Another proof has been given (p. 171), which depends upon the definition only. Prop. II., Chapter IV., viz. that the sum of the focal distances of a point on the ellipse is constant, has been proved without assuming the no less difficult proposition that every ellipse has two directrices. The Lemma being assumed, these results follow as in Chapter II. It is proved conversely, in the Appendix, that an ellipse, defined by the constant sum of its focal distances, may be generated in two ways by means of a focus and directrix. Prop. XIII., Chapter IV., is new, whilst Prop. VII. has been introduced as an important result admitting of a simple geometrical proof. Prop. II., in the second Chapter upon the hyperbola, is also new, and has been applied to prove, amongst other theorems, that the portion of any tangent intercepted between the asymptotes is bisected at the point of contact. I have in general made it an object to prove analogous properties by similar methods, the tendency of this arrangement being to diminish the labour of the student. Compare Props. XV., XVI., Chapter IV.; Props. XIII., XIV., Chapter VI. In the Chapter on Corresponding Points, the results of Orthogonal Projection are obtained by a method not involving solid geometry. The connection between these methods is shown at the end of Chapter XV. In Chapter XIII., the fundamental Anharmonic Properties of Conics are proved by general methods which were first exhibited, in the Quarterly Journal of Pure and Applied Mathematics, by Mr. B. W. Horne, Fellow of St. John's College. The principal properties of Poles and Polars are proved, in Chapter XIV., by methods applicable to all Conics. They are also proved for the parabola in Chapter III., and for central Conics in Chapter V. In the last Chapter, the method of Conical Projection has been explained and illustrated with the help of figures. The treatment of the subject is elementary and geometrical, no allusion being made to the analytical conception of imaginary points. The definitions are in a majority of instances placed at the beginnings of the various Chapters; some of the most general are given at the commencement of the work, whilst another class, in which especial explanation is required, may be found by referring to the Table of Contents. In references, the number of the page has usually been given, except when the proposition referred to occurs in the same Chapter as the reference. The symbol of equality has been used in stating proportions, as well adapted to express that equality or similarity of ratios by which proportion is defined, and as superior in distinctness to the symbol (::) more commonly employed. The term eccentricity has been defined, and used as a convenient abbreviation throughout the work. The rectangles contained by the segments of any two chords are The circle through the foci and any point on the curve meets the If TP, TQ be tangents, ≤ STQ = HTP The foot of the focal perpendicular upon the tangent lies on the |