IN WHICH THE STRAIGHT LINE AND PLANE ARE DERIVED FROM PROPERTIES OF THE SPHERE. Being an attempt to get rid of Axioms and Postulates; and particularly to establish the Theory of Parallel Lines without recourse to any principle not grounded on TO WHICH IS ADDED AN APPENDIX Containing Notices of Methods at different times proposed for getting over the difficulty in the Twelfth Axiom of Euclid. 205 By a Member of the University of Cambridge. FOURTH EDITION. CANTOS Published by Robert HEWARD, 5, Wellington Street, Strand, London. Sold there, and by Ridgway, Piccadilly ; and GRANT, Cambridge. Printed by T. C. Hansard, 32, Paternoster Row, London. PREFACE TO THE FOURTH EDITION. In the preceding Editions endeavour had been made to get rid of Axioms, and particularly to establish the Theory of Parallel Lines without recourse to any principle not grounded on previous demonstration. On showing the results to some of the leading mathematicians at Cambridge, they replied by the remark, that they had always felt something to be more urgently wanted for the emendation of Geometry,—which was, information on the nature and construction of the straight line and plane. It had been stated, about the time when the circumstances were engrossing the attention of the public, that NAPOLÉON on his voyage from Egypt amused himself and staff with circular geometry. What circular geometry might be, could only be collected from the tradition, that the problem given by the future Emperor was 'to divide the circumference of a circle into four equal parts by means of circles only. But this sufficed to indicate, that the idea which had passed through the mind of that eminent practical geometer, was that in the properties of the circle, or still more probably of the sphere, might be discovered the elements of geometrical organization. The author had in consequence been led at different times to attempt the collecting of the conditions, under which figures of |