GEOMETRY.

BOOK I.

DEFINITIONS AND REMARKS.

1. Extension has three dimensions, length, breadth, and thickness.

Geometry is the science which has for its object:

1st. The measurement of extension; and 2dly, To discover, by means of such measurement, the properties and relations of geometrical figures.

2. A Point is that which has place, or position, but not magnitude.

3. A Line is length, without breadth or thickness.

4. A Straight Line is one which lies

in the same direction between any two of its points.

5. A Curve Line is one which changes

is direction at every point.

The word line when used alone, will designate a straight line; and the word curve, a curve line.

6. A Surface is that which has length and breadth, without height or thickness.

7. A Plane Surface is that which lies even throughout its whole extent, and with which a straight line, laid in any direction, will exactly coincide in its whole length.

8. A Curved Surface has length and breadth without thickness, and like a curve line is constantly changing its direction.

9. A Solid or Body is that which has length, breadth, and thickness. Length, breadth, and thickness, are called dimen

Definitions.

sions. Hence, a solid has three dimensions, a surface two, and a line one. A point has no dimensions, but position only 10. Geometry treats of lines, surfaces, and solids.

11. A Demonstration is a course of reasoning which establishes a truth.

12. An Hypothesis is a supposition on which a demonstration may be founded.

13. A Theorem is something to be proved by demonstration. 14. A Problem is something proposed to be done.

15. A Proposition is something proposed either to be done or demonstrated—and may be either a problem or a theorem. 16. A Corollary is an obvious consequence, deduced from something that has gone before.

17. A Scholium is a remark on one or more preceding propositions.

18. An Axiom is a self evident proposition.

OF ANGLES.

19. An Angle is the portion of a plane included between two straight lines which meet at a common point. The two straight lines are called the sides of the angle, and the common point of intersection, the vertex.

Thus, the part of the plane included. between AB and AC is called an angle:

AB and AC are its sides, and Aits vertex.

B

We may,

An angle is generally read, by placing the letter at the vertex in the middle. Thus, we say, the angle CAB. however, say simply, the angle A.

20. One line is said to be perpendicular to another when it inclines no more to the one side than to the other

Definitions.

The two angles formed are then equal to each other. Thus, if the line DB is perpendicular to AC, the angle DBA will be equal to DBC.

21. When two lines are perpendicular to each other, the angles which they form are called right angles. Thus, DBA and DBC are called right angles.

22. An acute angle is less than a right angle. Thus, DBC is an acute angle.

23. An obtuse angle is greater than a right angle. Thus, DBC is an obtuse angle.

24. The circumference of a circle is a curve line all the points of which are equally distant from a certain point within called the centre.

Thus, if all the points of the curve AEB are equally distant from the centre C, this curve will be the circumference of a circle. 25. Any portion of the circumference, as AED, is called an arc.

26. The diameter of a circle is a straight line passing through the centre and terminating at the circumference. Thus, ACB is a diameter.

27. One half of the circumference, as ACB is called a semicircumference; and one quarter of the circumference, as AC, is called a quadrant.

Definitions.

28. The circumference of a circle is used for the measurement of angles. For this purpose it is divided into 360 equal parts called degrees, each degree into 60 equal parts called ninutes, and each minute into 60 equal parts called seconds. The degrees, minutes, and seconds are marked thus ° '"; and 9° 18′ 16′′, are read, 9 degrees 18 minutes and 16 seconds.

29. Let us suppose the circumference of a circle to be divided into 360 degrees, beginning at the point B. If through the point of division marked 40, we draw CE, then, the angle ECB will be equal to 40 degrees. If CF were drawn through

180

90 80

40

B

C

the point of division marked 80, the angle BCF would be equal to 80 degrees. X

OF LINES.

30. Two straight lines are said to be parallel, when being produced either way, as far as we please, they will not meet each other.

31. Two curves are said to be parallel or concentric, when they are the same distance from each other at every point.

32. Oblique lines are those which approach each other, and meet if sufficiently produced.

33. Lines which are parallel to the horizon, or to the water level, are called horizontal lines.

34. Lines which are perpendicular to the horizon, or to the water level, are called vertical lines.

Definitions.

OF PLANE FIGURES.

35. A Plane Figure is a portion of a plane terminated on all sides by lines, either straight or curved.

36. If the lines which bound a figure are straight, the space which they inclose is called a rectilineal figure, or polygon. The lines themselves, taken together, are called the perimeter of the polygon. Hence, the perimeter of a polygon is the sum of all its sides.

37. A polygon of three sides is called a triangle.

38. A polygon of four sides is called a quadrilateral.

39. A polygon of five sides is called a pentagon.

40. A polygon of six sides is called

■ hexagon.

41. A polygon of seven sides is called a heptagon. 42. A polygon of eight sides is called an octagon.