DEFINITIONS OF MATHEMATICAL TERMS.

30. A Demonstration, or Proof, is a course of reasoning by which the truth of a statement is deduced.

31. An Axiom is a statement of a truth which is selfevident.

32. A Theorem is a statement of a truth which is to be demonstrated.

33. A Problem is a statement of something to be done.

34. A Postulate is a problem whose solution is self-evident. 35. Axioms, theorems, and problems are called Propo

sitions.

36. A Corollary is a statement of a truth which is a direct inference from a proposition.

37. An Hypothesis is a supposition made in a proposition or in a demonstration.

38. A Scholium is a comment on one or more propositions.

39. AXIOMS.

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

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

3. If equals are subtracted from equals, the remainders are equal.

4. If equals are added to unequals, the sums are unequal. 5. If equals are subtracted from unequals, the remainders are unequal.

6. If equals are multiplied by equals, the products are equal.

7. If equals are divided by equals, the quotients are equal. 8. The whole is greater than any of its parts.

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

10. Only one straight line can join two points.

11. A straight line is the shortest distance from one point to another.

12. All right angles are equal.

40. POSTULATES.

1. A straight line can be drawn joining any two points. 2. A straight line can be produced to any length.

3. From the greater of two straight lines, a part can be cut equal to the less.

4. In a plane a circumference of a circle can be described, with any point as a centre, and any distance as a radius. 5. Figures can be freely moved in space without change of form or size.

41. DEMONSTRATION.

A Demonstration is a logical process, the premises being definitions and self-evident and previously established truths. There are two methods of demonstration, called the Direct Method and the Indirect Method.

The Direct Method proves a truth by referring to definitions and self-evident and previously deduced truths, and concludes directly with the proof of the truth in question.

The Indirect Method proves a truth by showing that a supposition of its falsity leads to an absurdity;-called also reductio ad absurdum.

Pupils frequently fall into errors of demonstration. Notable among these errors are Begging the Question and Reasoning in a Circle.

Begging the Question is a form of argument in which the truth to be proved is assumed as established.

Reasoning in a Circle is a form of argument in which a truth is employed to prove another truth on which the former depends for its proof.

Q.E. F. "

hypothesis.

quod erat demonstrandum (which was to

be demonstrated).

quod erat faciendum (which was to be done).

PERPENDICULAR AND OBLIQUE STRAIGHT LINES.

THEOREM I.

43. At a point in a straight line, only one perpendicular can be erected to that line.

To

A

prove that only one can be erected to AB at C.

Draw the oblique line CD.

Revolve CD about C so as to increase a and decrease LACD.

It is evident that in one position of CD, as EC, the adjacents are equal.

But then EC is to AB.

(20)

And there can be only one position of the line in which the adjacents are equal;

... only one can be erected to AB at the point C. Q. E. D.

THEOREM II.

44. The sum of the two adjacent angles formed by two lines which meet equals two right angles.

Let s a and ACD be formed by the line CD meeting AB.

45. COR. 1.-If one of the two adjacent angles formed by two lines which meet is a right angle, the other is also a right angle.

46. COR. 2.-The sum of all the angles formed at a common point on the same side of a straight line equals two right angles.