30. A BASE of a polygon is the side on which the polygon is supposed to stand. But in the case of the isosceles triangle, it is usual to consider that side the base which is not equal to either of the other sides.

31. An equilateral polygon is one which has all its sides equal. An equiangular polygon is one which has

all its angles equal. A regular polygon is one which is equilateral and equiangular.

32. Two polygons are mutually equilateral, when all the sides of the one equal the corresponding sides of the other, each to each, and are placed in the same order.

Two polygons are mutually equiangular, when all the angles of the one equal the corresponding angles of the other, each to each, and are placed in the same order.

33. The corresponding equal sides, or equal angles, of polygons mutually equilateral, or mutually equiangular, are called homologous sides or angles.

AXIOMS.

34. An AXIOM is a self-evident truth; such as,

1. Things which are equal to the same thing, are equal to each other.

2. If equals be added to equals, the sums will be equal. 3. If equals be taken from equals, the remainders will be equal.

4. If equals be added to unequals, the sums will be unequal.

5. If equals be taken from unequals, the remainders will be unequal.

6. Things which are double of the same thing, or of equal things, are equal to each other.

7. Things which are halves of the same thing, or of equal things, are equal to each other.

8. The whole is greater than any of its parts.

9. The whole is equal to the sum of all its parts.

10. A straight line is the shortest line that can be drawn from one point to another.

11. From one point to another only one straight line can be drawn.

12. Through the same point only one parallel to a straight line can be drawn.

13. All right angles are equal to one another.

14. Magnitudes which coincide throughout their whole extent, are equal.

POSTULATES.

35. A POSTULATE is a self-evident problem; such as,

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

2. That a straight line may be produced to any length. 3. That a straight line may be drawn through a given point parallel to another straight line.

4. That a perpendicular to a given straight line may be drawn from a point either within or without the line. 5. That an angle may be described equal to any given angle.

PROPOSITIONS.

36. A DEMONSTRATION is a course of reasoning by which a truth becomes evident.

37. A PROPOSITION is something proposed to be demonstrated, or to be performed.

A proposition is said to be the converse of another, when the conclusion of the first is used as the supposition in the second.

38. A THEOREM is something to be demonstrated.

39. A PROBLEM is something to be performed.

40. A LEMMA is a proposition preparatory to the demonstration or solution of a succeeding proposition.

41. A COROLLARY is an obvious consequence deduced from one or more propositions.

42. A SCHOLIUM is a remark made upon one or more preceding propositions.

43. An HYPOTHESIS is a supposition, made either in the

enunciation of a proposition, or in the course of a demon

stration.

PROPOSITION I.-THEOREM.

44. The adjacent angles which one straight line makes by meeting another straight line, are together equal to two right angles.

Let the straight line DC meet A B, making the adjacent angles ACD, DCB; these angles together will be equal to two right angles.

A

E

D

C

B

From the point C suppose CE to be drawn perpendicular to AB; then the angles ACE and ECB will each be a right angle (Art. 15). But the angle A CD is composed of the right angle A CE and the angle ECD (Art. 34, Ax. 9), and the angles ECD and DCB compose the other right angle, ECB; hence the angles ACD, DCB together equal two right angles.

45. Cor. 1. If one of the angles ACD, DCB is a right angle, the other must also be a right angle.

46. Cor. 2. All the successive angles, BA C, CAD, DAE, EAF, formed on the same side. of a straight line, BF, are equal, when taken together, to two right angles; for their sum is equal to that of the two adjacent angles, BAC, CAF.

D

C

E

B

F

A

PROPOSITION II.-THEOREM.

47. If one straight line meets two other straight lines at a common point, making adjacent angles, which together are equal to two right angles, the two lines form one and the same straight line.

If C B is not the straight line AC produced, let CE be that line produced; then the line ACE being straight, the sum of the angles ACD and DCE will be equal to two right angles (Prop. I.), But by hypothesis the angles ACD and DCB are together equal to two right angles; therefore the sum of the angles A CD and DCE must be equal to the sum of the angles ACD and DCB (Art. 34, Ax. 2). Take away the common angle A CD from each, and there will remain the angle D C B, equal to the angle DCE, a part to the whole, which is impossible; therefore CE is not the line A C produced. Hence AC and CB form one and the same straight line.

PROPOSITION III.-THEOREM.

48. Two straight lines, which have two points common, coincide with each other throughout their whole extent, and form one and the same straight line.

Let the two points which are common to two straight lines be

A and B.

F

A

BC

E

D

The two lines must coincide between the points A and B, for otherwise there would be two straight lines between A and B, which is impossible (Art. 34, Ax. 11).

Suppose, however, that, on being produced, the lines begin to separate at the point C, the one taking the direc