Sidebilder
PDF
ePub
[ocr errors][merged small]
[ocr errors]

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.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][subsumed][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

.

.

[ocr errors]

.

.

.

[ocr errors]

.

.

THE PRIMARY CONCEPTS OF GEOMETRY.

I. Definitions of Geometric Magnitudes.

ARTICLE

PAGE 66. Right angle defined.

13

34. Geometry defined

8 67. A perpendicular defined

13

35. A solid defined

8 68. Complement defined

13

36. A surface defined

8 69. Acute angle defined.

13

37. A line defined.

8 70. Obtuse angle defined

13

38. A point defined

8 71. Perigon defined

13

39. A magnitude defined

8 72. Sum of angles about a point is a

40-43. Point, line, surface, solid, de-

perigon.

14

fined, using motion .

8 73. Reflex angle defined

14

44. Why space is called tri-dimensional, 9 74. Oblique angles and lines defined

14

45. Straight line defined

9 75. Vertical angles defined.

14

46. A curve defined .

76. Bisector defined

14

47. Line henceforth means straight 77. A trace defined

14

line ..

9 78. A triangle defined

15

48. A sect defined .

9 79. Vertices defined

15

49. A plane defined

80. Sides defined

15

50. Plane defined, using motion

9 81. Interior angle defined

15

51. A figure defined .

82. Exterior angle defined .

15

52. A plane figure defined

9 83. Congruent defined

13

53. Circle and center defined

84. Equivalent defined

15

54. Radius defined

55. Diameter defined.

II. Properties of Distinct Things.

56. Arc defined .

57. Parallel lines defined

85-93. The so-called axioms of arith-

metic.

58. Angle, vertex, arms, defined .

59. Explement defined .

III. Some Geometrical Assumptions about

60. Equal angles defined

Euclid's Space.

61. Adjacent angles and their sum de-

fined ..

94-99. The so-called geometric axioms, 16

62. Straight angle defined .

IV. The Assumed Constructions.

63. When two adjacent angles equal a

straight angle .

12 100-103. The so-called postulates.

17

64. Supplement defined.

65. Supplemental adjacent angles 12 | Table of symbols used .

17

[ocr errors]
[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

ARTICLE

PAGE

PAGE

22

« ForrigeFortsett »