In America the geometries most in vogue at present are vitiated by the immediate assumption and misuse of that subtile term “ direction;" and teachers who know something of the Non-Euclidian, or even the modern synthetic geometries, are seeing the evils of this superficial “ directional” method. Moreover the attempt, in these books, to take away by definition from the familiar word “distance” its abstract character and connection with length-units, only confuses the ordinary student. A reference to the article Measurement in the “Encyclopædia Britannica” will show that around the word “distance” centers the most abstruse advance in pure An elementary geometry has no need of the words direction and distance. The present work, composed with special reference to use in teaching, yet strives to present the Elements of Geometry in a way so absolutely logical and compact, that they may be ready as rock-foundation for more advanced science and philosophy. 1 study. Besides the acquirement of facts, there properly belongs to Geometry an educational value beyond any other element ary subject. In it the mind first finds logic a practical instrument of real power. The method published in my Mensuration for the treatment of solid angles, with my words steregon and steradian, having been adopted by such eminent authorities, may I venture to recommend the use of the word sect suggested in the same volume ? From 1877 I regularly gave my classes the method of Book IX. In 1883 my pupil, H. B. Fine, at my suggestion, wrote out a Syllabus of Spherical Geometry on the lines of my teaching, which I have followed in Book IX. The figures, which I think give this geometry a special advantage, owe all their beauty to my colleague, Professor A. V. Lane, who has given them the benefit of his artistic skill and mastery of graphics. The whole work is greatly indebted to my pupil and friend, Dr. F. A. C. Perrine. We have striven after accuracy. Any corrections or suggestions relating to the book will be thankfully received. GEORGE BRUCE HALSTED. 2004 Matilde Street, Austin, Texas. . . . . . . . THE PRIMARY CONCEPTS OF GEOMETRY. I. Definitions of Geometric Magnitudes. 8 72. Sum of angles about a point is a 40-43. Point, line, surface, solid, de- 44. Why space is called tri-dimensional, 9 74. Oblique angles and lines defined 9 75. Vertical angles defined. 47. Line henceforth means straight 77. A trace defined 50. Plane defined, using motion II. Properties of Distinct Things. 85-93. The so-called axioms of arith- 58. Angle, vertex, arms, defined . III. Some Geometrical Assumptions about 61. Adjacent angles and their sum de- 94-99. The so-called geometric axioms, 16 IV. The Assumed Constructions. 63. When two adjacent angles equal a 12 100-103. The so-called postulates. 65. Supplemental adjacent angles 12 | Table of symbols used . ARTICLE PAGE PAGE PRIMARY RELATIONS OF LINES, ANGLES, AND TRIANGLES. II. Angles about Two Points. ARTICLE 104-105. Right and straight angles . 18, 19 11. Transversal defined ... 22 |