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GEOMETRY.

BOOK I.

THE PRINCIPLES.-DEFINITIONS.

1. GEOMETRY is that science which treats of Magnitudes. Magnitudes may be considered under three dimensions-length, breadth, thickness.

2. A line has length only, without breadth or thick

ness.

The extremities of a line are points; hence,

3. A point has neither length, breadth, nor thickness, but position only.

4. A straight line is one, the direction of which is always the same.

5. When two straight lines meet, the opening between them is called an angle; the point of meeting is called the vertex; and the lines themselves, which are said to contain the angle, are called the sides.

Thus the opening between

the lines AC, CB, is called

the angle made by those c

lines; the point at which C

is placed is called the vertex;

and the lines AC, BC, are called the sides.

An angle is sometimes designated by the letter at the vertex; or, more frequently, it is designated by the three letters at the extremities of the sides, the letter at the vertex being always placed in the middle of the three letters; thus the angle ACB, denotes the angle having the vertex C, and contained by the sides AC, BC.

The pupil should be careful not to confound an angle with the length of its sides. It is evident, for example, that the lines DE, FE, make a much greater angle, that

D

H

F

is, are much further apart, than the much longer lines GH, HI.

Angles, like all other quantities, are susceptible of being added to, and subtracted from each other: thus the angle ABD is the sum of the two angles ABC, CBD; and the angle ABC is the difference of the two

angles ABD, CBD.

B

B

6. When a straight line, as AB, meets another straight line, as CD, so as to make the adjacent angles, CAB, BAD, equal to each other, each of these angles is called a right angle, and the line AB is said to be perpendicular to CD.

A

7. Every angle less than a right angle is called an

[blocks in formation]

example, ABC is an acute, and DEF is an obtuse angle.

8. A plane is a surface, in which, if any two points be taken, the straight line drawn between those points will lie wholly in the surface.

9. Straight lines are parallel when they have the same direction, as AB, CD.

Parallel lines cannot meet, how far so ever they are produced, (that is, continued.)

D

10. A plane figure is a plane terminated on all sides by lines.

11. If the bounding lines are straight, the figure is called a polygon; and the lines themselves, taken together, form the contour, or perimeter of the figure.

12. Among polygons, are more particularly distinguished the figure of three sides, called a triangle, and that of four sides, called a quadrilateral.

13. An equilateral trian

gle is one which has all its sides equal.

14. An isosceles triangle

is one which has two equal sides.

15. When no two sides

are equal, the triangle is called scalene.

B

16. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypotenuse. Thus in the triangle ABC, right angled at A, BC is the hypotenuse.

17. The parallelogram is a quadrilateral whose opposite sides are parallel.

18. The trapezoid has only two of its opposite sides parallel.

19. The rhombus is a parallelogram whose sides are all equal.

20. A rectangle is a parallelogram having all its angles right angles.

21. A square is a rectangle having all its sides equal.

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