the position OC, continuous with the initial position AO, the angle thus formed is an angle equal to two right angles.

9. A reflex angle is that which is greater than two right angles.

Thus, when the line OB, in revolving round O, takes the position OD or OE, it forms with CAO two angles AOD and AOE, each of which is greater than two right angles. Such an angle is called a reflex angle.

D

10. An obtuse angle is that which is greater than a right angle, but less than two right angles.

11. An acute angle is that which is less than a right angle.

12. A figure is a portion of space enclosed by one or more boundaries.

13. A plane figure is that which is enclosed by one or more lines; if the lines are straight, the figure is called rectilineal. A solid figure is that which is enclosed by one or more surfaces.

14. A triangle is a plane figure contained by three straight lines.

Any one of the sides of a triangle in reference to the opposite angle, and in distinction from the other two sides, is called the base; the other two sides in reference to the base are often called the legs;" but the term is scarcely admissible, unless the two sides to which it applies be equal.

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15. A quadrilateral is a plane figure contained by four straight lines.

16. A polygon is a plane figure contained by more than four straight lines.

Schol.-A polygon is said to be regular when all its sides and angles are equal. It is said to be convex when no one of its angles is reflex.

Polygons are distinguished by names compounded of the Greek numerals and the Greek word for an angle. Thus, a figure of five sides is called a pentagon, and the others are hexagon, heptagon, octagon, nonagon, decagon, etc.

17. In a figure of four or more sides, a straight line joining the vertices of two angles not adjacent is called a diagonal.

18. An equilateral triangle is that which has its three sides equal.

19. An isosceles triangle is that which has two sides equal. 20. A scalene triangle is that which has all its sides unequal.

21. A right-angled triangle is that which has a right angle. The side opposite to, or subtending, the right angle is called the hypotenuse.

22. An obtuse-angled triangle is that which has an obtuse angle.

23. An acute-angled triangle is that which has all its angles acute.

This, and the two preceding definitions, imply the truth afterwards established, that every triangle has at least two acute angles. Triangles not having a right angle are often called oblique-angled triangles. The entire length or sum of the sides of a right lined figure is called the perimeter of the figure. The amount of surface contained within the boundaries of any plane figure is called its area.

24. Parallel straight lines are those which are on the same plane, and, being produced ever so far both ways, do not meet.

Parallel lines are manifestly equidistant at all parts, and a moment's consideration will shew that two straight lines on different planes, although they are not parallel, may yet never meet (see Prop. 14, sch.).

25. A parallelogram is a four-sided figure, whose opposite sides are parallel.

26. Any other quadrilateral is called a trapezium.

27. A rectangle is a parallelogram which has a right angle; it is said to be contained by any two adjacent sides.

This name is an abbreviation for "rectangular parallelogram." It might have been better that some name less liable to be misunderstood had been devised. It is sometimes called an oblong. Euclid uses the latter name in his definitions; elsewhere he uses the term rectangle.

28. A square is a rectangle whose two adjacent sides are equal.

29. A rhombus is a parallelogram whose two adjacent sides are equal, but its angles not right angles.

30. A circle is a plane figure contained by one line called the circumference; and is such that all straight lines drawn from a certain point within it to the circumference are equal to one another. This point is called the centre of the

circle.

31. An arc of a circle is any part of the circumference. 32. A chord is a straight line joining the two extremities of an arc.

33. A segment of a circle is the figure contained by an arc and its chord. The chord is called the base of the segment.

34. A diameter of a circle is a chord which passes through the centre. The half of a diameter is called a radius.

Cor.-It follows from the definition of a circle

1. That every point within the circle is at a distance from the centre less than the radius, and every point without the circle is at a distance from the centre greater than the radius.

2. That two circles are equal when the radius of the one is equal to the radius of the other, or when they have the same radius; for if the one circle be applied to the other, so that their centres coincide, their circumferences will coincide, since all the points of both are at the same distance from the centre.

35. A semicircle is a segment whose chord is a diameter. 36. A sector of a circle is the figure contained by any arc and two radii drawn through its extremities.

37. When the two radii of a sector are at right angles to one another, the sector is called a quadrant.

38. The altitude of any figure is the straight line drawn from its vertex perpendicular to its base.

So also are two angles

39. Two arcs, whose sum is the arc of a semicircle, are called supplements of one another. whose sum is two right angles.

40. Two arcs, whose sum is the called complements of one another. whose sum is a right angle.

arc of a quadrant, are So also are two angles

The arc of every quadrant is supposed to be divided into ninety equal parts, called degrees; and the right angle subtended by this arc is hence called an angle of ninety degrees.

Let it be granted

POSTULATES.

1. That a straight line may be drawn from any one point to any other point.

2. That a terminated straight line may be produced to any length in a straight line.

3. That a circle may be described from any centre, with a radius equal to any finite straight line.

AXIOMS.

1. Things which are equal to the same, or to equals, are equal to one another.

B

Thus, if we know that the line B is equal to the line A, and also that the line C is equal to the same line A, we must at once admit that the line B is equal to the line C; and from this it follows that the three lines are all equal.

2. If equals be added to equals the wholes are equal.

3. If equals be taken from equals the remainders are equal

4. If equals be added to unequals the wholes are unequal. 5. If equals be taken from unequals the remainders are unequal.

6. Things which are doubles of the same, or of equals, are equal to one another.

7. Things which are halves of the same, or of equals, are equal to one another.

8. Magnitudes which coincide, that is, which exactly fill the same space, are equal to one another.

9. The whole is greater than its part.

10. The whole is equal to all its parts taken together. 11. All right angles are equal to one another.

12. Two straight lines which intersect one another cannot be both parallel to the same straight line.

a. A straight line is defined by Euclid to be "that which lies evenly between its extreme points." But the word "evenly" seems to require definition as much as the word "straight." Every one, however, has a distinct notion of what a straight line is, and in what respect it differs from a curved or a crooked line; and no definition can convey a clearer idea.

b. A surface is plane when a rule, or straight edge, coincides with it in all positions. By a level surface an engineer means a portion of the surface of the terrestrial sphere, or one parallel to it, without inequalities, such as the surface of the sea, of a lake, or of the water in a canal.

c. A problem is something set forth to be done, such as to construct an equilateral triangle on a given line; to describe a circle which shall pass through three given points, etc. A theorem is something set forth to be demonstrated as true, such as that every triangle must have two acute angles; that all the angles of a triangle make up exactly two right angles, etc. A problem refers to the doing of something, a theorem to proving something. Proposition is a general term including both; it is a statement of something to be set before the mind. A lemma is a simple proposition preliminary to another of more importance which follows, and whose proof, in part, depends upon it. A corollary is an inference which follows immediately from some principle, statement, or proof going before. A scholium is a note or remark offered by way of explanation. In a theorem, something is always assumed or laid down as true, from which the proof is to be derived, or on which it is to be built up. This is called the hypothesis, or supposition; that which is to be proved is the conclusion or thesis. In a problem, that which is given is called the datum or data. Every problem consists of four parts(1) a general enunciation, (2) a particular enunciation, (3) a construc