In Section III. I have treated of Dihedral and Solid Angles, and in Section IV. of Polyhedra, and in particular of the five Regular Solids, with the definitions and a summary of the leading characteristics of the Prism, Cylinder, Pyramid, Cone, and Sphere. In Section V. the Volumes of Solids are considered, and the usual rules or formulæ for the numerical measures of volumes in cubic units deduced. To this Section I have added an Appendix on Symmetry, which I hope will be useful in assisting the student to clearer conceptions of figures and their relations in solid space. A short word has long been wanted for the decasyllabic "rectangular parallelepiped": I am sanguine that the word "cuboid," which I have here introduced, will meet with acceptance from mathematicians. In the concluding Section (VI.) I have given a sketch of the leading elementary propositions of Spherical Surface Geometry, partly as throwing light on, and suggesting convenient modes of treatment of problems in Solid Geometry, and as leading up to Spherical Trigonometry, but mainly with a view to a comparison with corresponding propositions in Plane Geometry. I have prefixed to the whole a "Preliminary Discussion on the Postulates of Geometry." It is in my opinion highly desirable that the student should at some point in his course examine carefully into the fundamental assumptions of Geometry, and the logical affiliation of its propositions. This is clearly impossible till he has attained to some familiarity with the subject, and it can hardly be done satisfactorily till he is prepared to enter on the study of Solid Geometry; but he may, and probably in most cases will, postpone it till a later period accordingly I have made the rest of the book quite independent of this discussion, which may be read or omitted by the student without prejudice to his clear understanding of the propositions in the Sections which follow. It will be seen that I have, in accordance with my title, strictly confined myself to the Elements of Solid Geometry, to the exclusion of the higher methods and results of Modern Geometry. These may best be studied in separate treatises, such as the "Projective Geometry" of Cremona, rather than in detached fragments introduced into a book whose main method is Euclidean, but to such treatises I have hopes that the present may form a useful introduction. I may add, in conclusion, that this treatise has been developed out of a "Syllabus of Solid Geometry," which I submitted to a Committee of the Association for the Improvement of Geometrical Teaching, and which that Committee received with a considerable amount of favour. Finding that it was difficult to make clear the intended thread of connection in the Syllabus by the bare enunciation of the Theorems in a certain sequence, I was led to write out the demonstrations, and then it seemed to me desirable to go further than I had originally proposed, and make a treatise, which might be considered as fairly complete, within the limits of Elementary Geometry, for the young student. In making this statement I must not be understood to claim for my work that it is issued with the sanction or authority of the Association or of the Committee referred to above. I am alone responsible for the entire contents, though I gratefully acknowledge that I have profited by the criticisms on my Syllabus of certain members of the Committee. HARROW, April, 1890. R. B. H. TABLE OF CONTENTS. PAGES PRELIMINARY DISCUSSION OF THE POSTULATES OF GEOMETRY I to 12 ERRATA. P. 29.-Reference, for "Th. VII." read "Th. VI." P. 30.-Last line, for "Th. I." read "Th. VIII." |