GENERAL TERMS.

18. A proof is a course of reasoning by which the truth or falsity of any statement is logically established.

19. An axiom is a statement admitted to be true without proof.

20. A theorem is a statement to be proved.

21. A construction is the representation of a required figure by means of points and lines.

22. A postulate is a construction admitted to be possible.

23. A problem is a construction to be made so that it shall satisfy certain given conditions.

24. A proposition is an axiom, a theorem, a postulate, or a problem.

25. A corollary is a truth that is easily deduced from known truths.

26. A scholium is a remark upon some particular feature of a proposition.

27. The solution of a problem consists of four parts:

1. The analysis, or course of thought by which the construction of the required figure is discovered.

2. The construction of the figure with the aid of ruler and compasses.

3. The proof that the figure satisfies all the conditions.

4. The discussion of the limitations, if any, within which the solution is possible.

28. A theorem consists of two parts: the hypothesis, or that which is assumed; and the conclusion, or that which is asserted to follow from the hypothesis.

29. The contradictory of a theorem is a theorem which must be true if the given theorem is false, and must be false if the given theorem is true. Thus,

A theorem:

If A is B, then C is D.

Its contradictory: If A is B, then C is not D.

30. The opposite of a theorem is obtained by making both the hypothesis and the conclusion negative. Thus,

A theorem:

Its opposite:

If A is B, then C is D.

If A is not B, then C is not D.

31. The converse of a theorem is obtained by interchanging the hypothesis and conclusion. Thus,

A theorem:

Its converse:

If A is B, then C is D.

If C is D, then A is B.

32. The converse of a truth is not necessarily true.

Thus, Every horse is a quadruped is true, but the converse, Every quadruped is a horse, is not true.

33. If a direct proposition and its opposite are true, the converse proposition is true; and if a direct proposition and its converse are true, the opposite proposition is true.

Thus, if it were true that

1. If an animal is a horse, the animal is a quadruped;

2. If an animal is not a horse, the animal is not a quadruped; it would follow that

3. If an animal is a quadruped, the animal is a horse.

Moreover, if 1 and 3 were true, then 2 would be true.

1. Magnitudes which are equal to the same magnitude, or equal magnitudes, are equal to each other.

2. If equals are added to equals, the sums are equal.

3. If equals are taken from equals, the remainders are equal. 4. If equals are added to unequals, the sums are unequal in the same order; if unequals are added to unequals in the same order, the sums are unequal in that order.

5. If equals are taken from unequals, the remainders are unequal in the same order; if unequals are taken from equals, the remainders are unequal in the reverse order.

6. The doubles of the same magnitude, or of equal magnitudes, are equal; and the doubles of unequals are unequal.

7. The halves of the same magnitude, or of equal magnitudes, are equal; and the halves of unequals are unequal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts.

Q.E.D. stands for quod erat demonstrandum, which was to be proved.
Q.E.F. stands for quod erat faciendum, which was to be done.
The signs +,
−, ×, ÷, =, have the same meaning as in Algebra.

PLANE GEOMETRY.

Book I.

RECTILINEAR FIGURES.

DEFINITIONS.

36. A straight line is a line such that any part of it, however placed on any other part, will lie wholly in that part if its extremities lie in that part, as AB.

37. A curved line is a line no part of A

which is straight, as CD.

38. A broken line is made up of dif- Eferent straight lines, as EF.

NOTE. A straight line is often called simply a line.

FIG. 4.

B

F

39. A plane surface, or a plane, is a surface in which, if any two points are taken, the straight line joining these points lies wholly in the surface.

40. A curved surface is a surface no part of which is plane.

41. A plane figure is a figure all points of which are in the same plane.

42. Plane figures which are bounded by straight lines are called rectilinear figures; by curved lines, curvilinear figures.

43. Figures that have the same shape are called similar. Figures that have the same size but not the same shape are called equivalent. Figures that have the same shape and the same size are called equal or congruent.

7

THE STRAIGHT LINE.

44. Postulate. A straight line can be drawn from one point to another.

45. Postulate. A straight line can be produced indefinitely.

46. Axiom.* Only one straight line can be drawn from one point to another. Hence, two points determine a straight line.

47. COR. 1. Two straight lines which have two points in common coincide and form but one line.

48. COR. 2. Two straight lines can intersect in only one point.

For if they had two points common, they would coincide and not intersect.

Hence, two intersecting lines determine a point.

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

50. DEF. The distance between two points is the length of the straight line that joins them.

51. A straight line determined by two points may be considered as prolonged indefinitely.

52. If only the part of the line between two fixed points is considered, this part is called a segment of the line.

53. For brevity, we say "the line AB," to designate a segment of a line limited by the points A and B.

54. If a line is considered as extending from a fixed point, this point is called the origin of the line.

*The general axioms on page 6 apply to all magnitudes. Special geometrical axioms will be given when required.