30. An Acute-Angled Triangle is one in which each angle is acute.

31. An Equiangular Triangle is one

having its three angles equal.

Equiangular triangles are also equilateral, and vice versa.

FIGURES OF FOUR SIDES.

32. A Quadrilateral is a polygon having four sides and four angles.

33. A Parallelogram is a quadrilateral which has its opposite sides parallel.

Parallelograms are denominated from the relations both of their sides and angles.

34. A Rectangle is a parallelogram having its angles right angles.

35. A Square is an equilateral rectangle.

36. A Rhomboid is an oblique-angled parallelogram.

37. A Rhombus is an equilateral rhomboid.

38. A Trapezium is a quadrilateral having no two sides parallel.

39. A Trapezoid is a quadrilateral in which two opposite sides are parallel, and the other two oblique.

40. Polygons bounded by a greater number of sides

than four are denominated only by the number of sides. A polygon of five sides is called a Pentagon; of six, a Hexagon; of seven, a Heptagon; of eight, an Octagon; of nine, a Nonagon, etc.

41. Diagonals of a polygon are lines

joining the vertices of angles not adjacent.

42. The Perimeter of a polygon is its boundary consid ered as a whole.

43. The Base of a polygon is the side upon which the polygon is supposed to stand.

44. The Altitude of a polygon is the perpendicular distance between the base and a side or angle opposite the base.

45. Equal Magnitudes are those which are not only equal in all their parts, but which also, when applied the one to the other, will coincide throughout their whole

extent.

46. Equivalent Magnitudes are those which, though they do not admit of coincidence when applied the one to the other, still have common measures, and are therefore numerically equal.

47. Similar Figures have equal angles, and the same number of sides.

Polygons may be similar without being equal; that is, the angles and the number of sides may be equal, and the length of the sides and the size of the figures unequal.

THE CIRCLE.

48. A Circle is a plane figure bounded by one uniformly curved line, all of the points in which are at the same c distance from a certain point within, called the Center.

49. The Circumference of a circle is the curved line that bounds it.

B

A

50. The Diameter of a circle is a line passing through its center, and terminating at both ends in the circumference.

51. The Radius of a circle is a line extending from its center to any point in the circumference. It is one half of the diameter. All the diameters of a circle are equal, as are also all the radii.

52. An Arc of a circle is any portion of the circumference.

53. An angle having its vertex at the center of a circle is measured by the arc intercepted by its sides. Thus, the arc AB measures the angle AOB; and in general, to compare different angles, we have but to compare the arcs, included by their sides, of the equal circles having their centers at the vertices of the angles.

UNITS OF MEASURE.

54. The Numerical Expression of a Magnitude is a number expressing how many times it contains a magnitude of the same kind, and of known value, assumed as a unit. For lines, the measuring unit is any straight line of fixed value, as an inch, a foot, a rod, etc.; and for surfaces, the measuring unit is a square whose side may be any linear unit, as an inch, a foot, a mile, etc. The linear unit being arbitrary, the surface unit is equally so; and its selection is determined by considerations of convenience and propriety.

A

B

For example, the parallelogram ABDC is measured by the number of linear units in CD, multiplied by the number of linear units in AC or BD; the product is the square units in ABDC. For, conceive CD to be composed of number any of equal parts—say five-and each part some unit of linear measure, and AC composed of three such units; from each point of division on CD draw lines parallel to AC, and from each point of division on AC draw lines parallel to CD or AB; then it is as obvious

as an axiom that the parallelogram will contain 5 × 3 = 15 square units. Hence, to find the areas of right-angled parallelograms, multiply the base by the altitude.

EXPLANATION OF TERMS.

55. An Axiom is a self-evident truth, not only too simple to require, but too simple to admit of, demonstration. 56. A Proposition is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem.

57. A Problem is something proposed to be done.

58. A Theorem is something proposed to be demonstrated.

59. A Hypothesis is a supposition made with a view to draw from it some consequence which establishes the truth or falsehood of a proposition, or solves a problem.

60. A Lemma is something which is premised, or demonstrated, in order to render what follows more easy.

61. A Corollary is a consequent truth derived immediately from some preceding truth or demonstration. 62. A Scholium is a remark or observation made upon something going before it.

63. A Postulate is a problem, the solution of which is self-evident.

POSTULATES.

Let it be granted

I. That a straight line can be drawn from any one point to any other point;

II. That a straight line can be produced to any distance, or terminated at any point;

III. That the circumference of a circle can be described about any center, at any distance from that center.

AXIOMS.

1. Things which are equal to the same thing are equal to each other.

2. When equals are added to equals the wholes are equal. 3. When equals are taken from equals the remainders are equal.

4. When equals are added to unequals the wholes are unequal.

5. When equals are taken from unequals the remainders are unequal.

6. Things which are double of the same thing, or equal things, are equal to each other.

7. Things which are halves of the same thing, or of equal things, are equal to each other.

8. The whole is greater than any of its parts.

9. Every whole is equal to all its parts taken together. 10. Things which coincide, or fill the same space, are identical, or mutually equal in all their parts.

11. All right angles are equal to one another.

12. A straight line is the shortest distance between two points.

18. Two straight lines cannot inclose a space.

ABBREVIATIONS.

The common algebraic signs are used in this work, and demonstrations are sometimes made through the medium of equations; and it is so necessary that the student in geometry should understand some of the more simple operations of algebra, that we assume that he is acquainted with the use of the signs. As the terms circle, angle, triangle, hypothesis, axiom, theorem, corollary, and definition, are constantly occurring in a course of geometry, we shall abbreviate them as shown in the following list: