360 equal spaces about the point A, which is contained between

AB and AC.

Angles are distinguished in respect to magnitude by the terms Right, Acute, and Obtuse Angles.

15. A Right Angle is that formed by one line meeting another, so as to make equal angles with that other.

The lines forming a right angle are perpendicular to each other.

16. An Acute Angle is less than a right angle.

17. An Obtuse Angle is greater than a right angle.

Obtuse and acute angles are also called

oblique angles; and lines which are neither parallel nor perpendicular to each other are called oblique lines.

18. The Vertex or Apex of an angle is the point in which the including lines meet.

19. An angle is commonly designated by a letter at its vertex; but when two or more angles have their vertices at the same point, they cannot be

thus distinguished.

For example, when the three lines AB, AC, and AD meet in the common point A, we designate either of the angles formed, by three letters, placing that at the vertex between those at the opposite extremities of the including lines. Thus, we say, the angle BAC,

etc.

A

B

20. Complements. Two angles are said to be comple ments of each other, when their sum is equal to one right angle.

21. Supplements. - Two angles are said to be supplements of each other, when their sum is equal to two right angles.

PLANE FIGURES.

22. A Plane Figure, in geometry, is a portion of a plane bounded by straight or curved lines, or by both combined.

23. A Polygon is a plane figure bounded by straight lines, called the sides of the polygon.

The least number of sides that can bound a polygon is three, and by the figure thus bounded all other polygons are analyzed.

FIGURES OF THREE SIDES.

24. A Triangle is a polygon having three sides and three angles.

Tri is a Latin prefix signifying three; hence a Triangle is literally a figure containing three angles. Triangles are denominated from the relations both of their sides and angles.

25. A Scalene Triangle is one in which no two sides are equal.

26. An Isosceles Triangle is one in which two of the sides are equal.

27. An Equilateral Triangle is one in which the three sides are equal.

28. A Right-Angled Triangle is one which has one of the angles a right angle.

29. An Obtuse-Angled Triangle is one aving an obtuse angle.

A

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

31. An Equiangular Triangle is one

having its three angles qual.

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 rela

tions 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 rhomDoid.

38. A Trapezium is a quadrilateral having 10 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, I 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 througn 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.

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 any number A

B

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