AN ELEMENTARY COURSE OF MATHEMATICS FOR THE USE OF THE ROYAL MILITARY ACADEMY, AND FOR STUDENTS IN GENERAL. BY S. HUNTER CHRISTIE, M.A. OF TRINITY COLLEGE, CAMBRIDGE, SECRETARY OF THE ROYAL SOCIETY, VICE-PRESIDENT OF THE ROYAL ASTRONOMICAL SOCIETY, PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY. VOL. II. PART III. GEOMETRY. Published by the Authority of THE MASTER-GENERAL AND THE BOARD OF ORDNANCE. PREFACE TO PART III. THIS Part of the Course of Mathematics for the Royal Military Academy contains a Supplement to the first six books of Euclid; the Elements of the Geometry of Planes and of Solids; the elementary principles of Descriptive Geometry; and the principles of Isometric Perspective. It was originally my intention that this Part should contain the Elements of Plane Geometry with the Geometry of Planes and of Solids, but having been called upon to furnish a short treatise on the elementary principles of Descriptive Geometry, adapted to the wants of this Institution, I was desirous of avoiding the delay which would arise from commencing with Plane Geometry. When the greater portion of this Part of the Course was printed, and had for some time been in use in the Academy, a new edition of Euclid's elements, by Mr. Robert Potts, M.A. of Trinity College, Cambridge, which is likely to supersede most others, to the extent, at least, of the six books, was published. From the manner of arranging the demonstration, this edition has the advantages of the symbolical form, and it is at the same time free from the manifold objections to which that form is open. The duodecimo edition of this work, comprising only the first six books of Euclid, with Deductions from them, having been introduced at this institution as a text book, now renders any other treatise on Plane Geometry unnecessary in our Course of Mathematics. It has however been found necessary to have a Supplement to these elements, in order to establish the principles on which problems on the mensuration of the circle depend, and this has been made to precede the Geometry of Planes. Besides the propositions in the eleventh book of Euclid, there are several propositions on straight lines and planes which are of great use, particularly as properties to refer to in problems on projections and Descriptive Geometry. These have been added in the Geometry of Planes. To the Geometry of Solids we must 'refer for all the principles requisite for obtaining rules for the mensuration of solids. There is here little difficulty until we come to the pyramid, the cylinder, the cone and the sphere. It is true that the Gordian knot which is here presented may be cut by adopting the method of Cavalieri, but besides that this method is erroneous in principle, no conclusion can be drawn from it in the case of the simplest solids, without, in fact, assuming the very thing to be proved it ought therefore never to be had recourse to in sound elementary instruction. The method which, after much reflection on the subject, I considered the simplest, and at the same time the most satisfactory, differed but little from that adopted by Playfair in the last ten propositions in the third book. of the Supplement to his Elements of Geometry. I have therefore here followed his method though I have not always adhered to the form of his demonstrations. Little that is original can be expected in any elementary treatise on Descriptive Geometry. In Monge, Hachette or Vallée, I had ample materials for much more than I proposed giving, but I considered that my object would be best attained by taking as a guide an elementary treatise which had been found most practically useful in the country which gave birth to this branch of Geometry, and in which it has been most extensively cultivated and applied. Lacroix's elementary treatise, which was the first published, with the exception of Monge's Leçons, is still one of the best to the extent I proposed to go into the subject, and is well adapted for instruction; but on comparing it with Lefebure de Fourcy's I found that the latter was best calculated for the object I had in view. In the order of the Problems I have nearly followed De Fourcy, but I have endeavoured to present the principles of the science, and the construction of the Problems, in a form that should obviate the difficulties which may present themselves to a learner on his first entering on this subject. I have also added several problems not given in De Fourcy, the constructions of some of which are, as far as I know, original. The explanations which are here given of the principles of Descriptive Geometry and their application to most of the problems on Planes and Right lines will, I hope, enable the student to take up the subject, at the point at which it has been left, in De Fourcy or some of the more extensive works to which I have referred, and to pursue it to any extent that circumstances may render necessary. In the Geometry of Planes I have likewise followed |