9. One line is Perpendicular to another when the first inclines not more toward the second on the one side than on the other.

10. An Angle is the difference of direction of two lines.*

The point where the two lines meet is called the vertex of the angle.

11. Angles are Right or Oblique.

12. A Right Angle is that which is made by one line perpendicular to another.

Or, when the angles on either side of one line meeting another are equal, they are right angles.

13. Oblique Angles are either Acute or Obtuse.

14. An Acute Angle is less than a right

angle.

15. An Obtuse Angle is greater

than a right angle.

* When one line, having comcided with another, begins to move round the point at one extremity, it begins to have a different direction, and the amount of this difference depends upon the amount of the movement, which is evidently measured by the portion of the circumference described by the other extremity. The length of this portion is usually expressed in degrees, each degree being the part of the whole circumference.

th

It is immaterial whether the revolving line be longer or shorter, when it has attained the same position with respect to the stationary line, or the same difference of direction from it, the portion of circumference described will contain the same number of degrees in both circumferences; each degree in the smaller circumference being smaller, since it is the 360th part of its own circumference.

Parallel lines having no difference of direction, the angle which they make with each other is zero or 0°.

Fractions of a degree are expressed usually in minutes and seconds, a minute being the 60th part of a degree, and a second the 60th part of a minute. This is called the sexagesimal measurement of angles, or division of the circumference. Another mode of division sometimes used, is of the whole circumference into four parts called quadrants, the quadrant into 100 parts called grades, or centesimal degrees, each grade into 100 centesimal minutes, and each minute into 100 centesimal seconds. This is called the centesimal division of the circumference. To convert one kind of degree into the other, it is only necessary to observe that a grade is 0-9 of a degree.

An angle is named from the letter at its vertex. Thus we say the angle A. When, however, there are two angles whose vertices are at the same point, this method would be ambiguous. It is necessary, then, to designate the angle to be A4 pointed out by three letters, naming the one at the vertex always in the middle. Thus, the angle formed by the two lines CB and CE is called the angle BCE, or ECB; and the angle formed by the two lines CE and CD is called the angle ECD, or DCE.

B

E

D

Angles are susceptible of addition, subtraction, and multiplication. Thus the angle BCD BCE+ECD.

16. Superficies are either Plane or Curved.

17. A Plane Superficies, or a Plane, is that which is straight in every direction, or with which a right line, joining any two points of it, will coincide throughout the length of the line. But if not, it is curved.

18. More accurately, a Curve Surface is one of which the section made by some plane cutting it is

a curve.*

19. A Plane Figure is a portion of a plane, bounded either by right lines or curves.

20. Plane figures that are bounded by right lines are called Polygons, and have names according to the number of their sides, or of their angles; the number of sides and angles being the same. The least number of sides requisite to form a polygon is

three.

21. A Polygon of three sides and three angles is called a Triangle. And it receives particular denominations from the relations of its sides and angles.

22. An Equilateral Triangle is one the three sides of which are all equal.

* A curve surface may or may not be straight in certain directions. See the cone and cylinder, toward the end of the volume.

23. An Isosceles Triangle is one which has two sides equal.

24. A Scalene Triangle is one whose three sides are all unequal.

25. A Right-angled Triangle is a triangle having one right angle.

It will be shown hereafter that no triangle can have more than one right angle, or more than one obtuse angle.

26. Other triangles are Oblique-angled, and are either obtuse or acute.

27. An Obtuse-angled Triangle has one obtuse angle. 28. An Acute-angled Triangle has its three angles

acute.

29. A figure of Four sides and angles is called a Quadrangle, or a Quadrilateral.

30. A Parallelogram is a quadrilateral which has both its pairs of opposite sides parallel. And it takes the following particular names, viz., Rectangle, Square, Rhombus, Rhomboid.

31. A Rectangle is a right-angled parallelogram.

32. A Square is an equilateral rectangle.

33. A Rhomboid is an oblique-angled parallelogram.

34. A Rhombus is an equilateral rhomboid.

35. A Trapezium is a quadrilateral which has not

its opposite sides parallel.

36. A Trapezoid is a quadrilateral which has only one pair of opposite

sides parallel.

37. A Pentagon is a polygon of five sides; a Hexagon is one of six sides; a Heptagon, one of seven; an Octagon, one of eight; a Nonagon, one of nine; a Decagon, one of ten; an Undecagon, one of eleven; and a Dodecagon, one of twelve sides.

The Perimeter of a polygon is the sum of its bounding lines.

A Convex Polygon is one the perimeter of which can be intersected by a straight line in but two points.

38. A Polygon is Equilateral when all its sides are equal; and it is Equiangular when all its angles are equal. A Regular Polygon is one which is both equiangular and equilateral.

39. An Equilateral Triangle is a regular polygon of three sides, and the square is one of four; the former being also called a trigon, and the latter a tetragon.

40. A Diagonal is a line joining any two angles of a polygon not adjacent.

41. A Circle is a plane figure bounded by a curve line, called the Circumference, every point of which is equidistant from a certain point within, called the Center.

42. The Radius of a circle is a line drawn from the center to the circumference.

43. The Diameter of a circle is a line drawn through the center, and terminating both ways at the circumference.

44. An Arc of a circle is any part of the circumference.*

* It will be shown hereafter that the circumference of a circle may be obtained by multiplying the radius by 6-2832: in order to obtain the absolute length of any arc given in degrees and parts of a degree, or grades and parts, it is necessary to ascertain what fraction of a

45. A Chord is a right line joining the extremities of an arc.

46. A Segment is any part of a circle bounded by an arc and its chord.

47. A Semicircle is half the circle, or a segment cut off by a diameter. A Semicircumference is half the circumference.

48. A Sector is a part of a circle which is bounded by an arc, and two radii.

Note.-A sector is a surface, as is also a segment.

49. A Quadrant, or Quarter of a circle, is a sector having a quarter of the circumference for its arc, its two radii being perpendicular to each other. A quarter of the circumference is also called a Quadrant.

Note.-A semicircle contains 180 degrees, and a quadrant 90 degrees.

50. Concentric Circles are those which have the same center.

51. Circles are said to be Eccentric with respect to one another when they have not the same center. In this case, the one circumference may be, with respect to the other, Exterior, Interior, Tangent Externally, Tangent Internally, or, finally, the two circumferences may intersect.

52. An Angle in a Segment is that which is contained by two lines, drawn from any point in the arc of the segment, to the two extremities of that arc. Thus A and D are both angles in the seg- B ment BADC. They are also called inscribed angles, and are said to be inscribed in the segment.

Ꭺ

E

D

circumference the arc is, by reducing 360° to the lowest denomination in the given arc for a denominator, and the degrees, &c., of the arc to the same denomination, for a numerator, then to multiply this fraction by the product of the radius and the number 6.2832.