ΤΟ WILLIAM CRACKANTHORPE, Esq., THIS BOOK IS INSCRIBED AS A TESTIMONY OF THE HIGHEST RESPECT AND ESTEEM, AND AS A GRATEFUL ACKNOWLEDGMENT OF NUMEROUS AND EXTENSIVE OBLIGATIONS CONFERRED UPON THE AUTHOR. PREFACE. THE Examination Papers which the Author has selected for solution in the present Treatise have been proposed in the several years from 1830 to 1846 to the students of St John's College at the end of their fourth term of residence, and according to the plan which he adopted in the solution of the Trigonometrical Problems, he has endeavoured to place them before the reader in a proper form for the inspection of the examiner. The problems are sufficiently varied in their character to exercise the student in the ordinary properties of the straight line, circle, and conic sections; they have been proposed by some of the most distinguished members of the society; the generality of the results are remarkable for their neatness and simplicity; and except in one instance it is needless here to make any further comment. In Question 6, Dec. 1833. (No. IV), it is required "To inscribe in a circle a triangle whose sides or sides produced shall pass through three given points in the same plane." This problem has been solved analytically in a most ingenious and elegant treatise, entitled "Researches on Curves of the second order" lately published by Mr Hearn, who remarks that it was proposed by M. Cramer to M. de Castillon, and that Lagrange has given a purely analytical solution which may be found in the memoirs of the Academy of Berlin (1776). The problem then being one of acknowledged difficulty, the author hopes that the first Appendix in which an analytical solution has been given when a conic section is substituted for the circle, will not be entirely devoid of interest. The case of the three different conic sections has been separately considered, and the author has afterwards still further generalized the problem, by inscribing in a given conic section, a polygon whose n sides taken in order shall pass through n fixed points. A simple and concise geometrical solution of M. Cramer's problem has been extracted from "The Liverpool Apollonius by J. H. Swale," and inserted in the third Appendix; and when the triangle is to be inscribed in a conic section, the problem has been reduced to that of inscribing in a circle a triangle whose three sides shall pass through three fixed points, so as to afford a comparatively simple geometrical solution in the more general case. Some apology may be considered necessary for the introduction into the second appendix of two different methods of determining the magnitude and position of the conic section represented by the general equation of the second degree. |