OF PLANE CURVES WITH NUMEROUS EXAMPLES BY T. H. EAGLES, M.A. INSTRUCTOR IN GEOMETRICAL DRAWING AND LECTURER IN ARCHITECTURE VU London: MACMILLAN AND CO. 1885 [All Rights reserved.] PREFACE. THE appearance of another text-book on Geometry may perhaps be considered to demand an apology, but I venture to hope that an examination of the following pages will shew them to differ considerably from any existing treatise. The extending use of graphic methods in the solution of many practical engineering problems has appeared to me to demand a corresponding extension in the practice of drawing the curves on which such solutions may frequently depend, and, though the properties of conic sections have been discussed thoroughly both geometrically and analytically, there is so far as I am aware no book treating of the actual delineation of the curves from given data to anything like the extent here attempted. Independently however of their applied use, the problems generally will, I think, be found useful merely as drawing exercises in science and other schools. A great deal of attention is devoted to the construction of regular polygons, circles packed into another circle and similar fancy figures, by methods which no practical draughtsman ever uses, while the construction of an ellipse is at the most limited to drawing it from the principal axes or from a pair of conjugate diameters; and the time spent on these and similar exercises might, I think, E. b be more profitably devoted to work bringing out the nature and properties of this and other curves. I can say from experience that the practice of sketching a curve freehand through a series of previously found points is a most valuable element in teaching mechanical drawing, while the finding the points furnishes abundant exercise in handling square and compasses, and impresses on the student in a very striking manner the necessity for neatness and accuracy in their use. Each problem may of course be drawn on paper without reference to the proof of the principle on which its construction depends, but I consider that for the advanced student at any rate it must be much more satisfactory to work with as complete an insight as possible into the methods he is using instead of groping along by mere rule of thumb, so that in nearly all cases notes in proof of the property made use of have been added, although such proofs may be found in numerous published works, and are indeed so completely common property that I have not thought it necessary to give direct references to the pages from which they have been taken. I cannot however here omit to notice my indebtedness to Dr Salmon's classical work on Conic Sections, or to Chasles' Géométrie Supérieure for the chapter on Anharmonic Ratio and the Anharmonic Properties of Conics. Chap. VIII. will, I hope, convince a draughtsman that he can if he likes make use of an engine very little known in England and of enormous power. The methods of Modern Geometry deserve to be brought into much closer relation with the drawing-board than has hitherto been the case. The chapter on Plane Sections of the Cone and Cylinder involves some elementary notions of Solid Geometry or Orthographic Projection, but the explanations given will, I hope, enable the average student to work through the chapter without referring to any special treatise on Projection. The ordinary pseudo-perspective diagrams usually given in books on Conics are I think unsatisfactory, and the method of referring the solid to two rectangular planes seems to me in every way preferable. When the mental conception of a plan and elevation is once thoroughly realised the student is well repaid by the exactness with which he is able to lay down on paper any point or line on the surface of the cone. The later chapters cannot be read without some knowledge of trigonometry, but the practice of translating a trigonometrical expression into something which can be represented to the eye is a valuable one, and the hints given in the chapter on the Graphic Solution of Equations will I trust be found useful. My warmest thanks are due to my friend and colleague Professor Minchin for much valuable advice and assistance most freely and readily given: without his help the book would have been much less complete than it is, whatever its imperfections may be found to be. It would be too much to hope that a work of this character should have been compiled and gone through the press without some errors creeping in. I hope they are not more numerous than from the nature of the case may be considered unavoidable, and I shall be thankful for any such being brought to my notice. COOPERS HILL, Oct. 1885. |