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34. Now since the angular magnitude about the point O is neither increased nor diminished by the number of lines which radiate from that point, the sum of all the angles about a point in a plane, as AOB+BOC+CO D, etc., in Fig. 1, is equal to four right angles; and the sum of all the angles about a point on one side of a straight line drawn through that point, as AOB+BOC+ COD, etc., Fig. 2, is equal to two right

angles.

Hence two adjacent angles, O CA and OC B, formed by two straight lines, of which one is produced from the point of meeting in both directions, are supplements of each other, and may A be called supplementary adjacent angles.

ON THE METHOD OF SUPERPOSITION.

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35. The test of the equality of two geometrical magnitudes is that they coincide point for point.

Thus, two straight lines are equal, if they can be so placed that the points at their extremities coincide. Two angles are equal, if they can be so placed that their vertices coincide in position and their sides in direction.

In applying this test of equality, we assume that a line may be moved from one place to another without altering its length ; that an angle may be taken up, turned over, and put down,

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C

CI

B'

angles, ABC and A'B'C'. Let B the side BC be placed on the side

B'C', so that the vertex B shall fall on B', then if the side BA fall on B'A', the angle ABC equals the angle A'B' C"; if the side BA fall between B'C' and B'A' in the direction B' D, the angle ABC is less than A'B'C"; but if the side B A fall in the direction B'E, the angle ABC is greater than A' B' C'.

This method of superposition enables us to add magnitudes of the same kind. Thus, if we have two straight lines A B and CD, by

A

C

A

BC

-D

D

B

placing the point C on B, and keeping CD in the same direction with AB, we shall have one continuous straight line A D equal to the sum of the lines A B

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36. DEF. A Demonstration is a course of reasoning by which the truth or falsity of a particular statement is logically established. 37. DEF. A Theorem is a truth to be demonstrated.

38. DEF. A Construction is a graphical representation of

a geometrical conception.

39. DEF. A Problem is a construction to be effected, or a question to be investigated.

1

3=

23

40. DEF. An Axiom is a truth which is admitted without demonstration.

41. DEF. A Postulate is a problem which is admitted to be possible.

42. DEF. A Proposition is either a theorem or a problem. 43. DEF. A Corollary is a truth easily deduced from the proposition to which it is attached.

44. DEF. A Scholium is a remark upon some particular feature of a proposition.

L

45. DEF. An Hypothesis is a supposition made in the L enunciation of a proposition, or in the course of a demonstration.

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46. AXIOMS.

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

2. When equals are added to equals the wholes are equal.

3. When equals are taken from equals the remainders are equal. 4. When equals are added to unequals the wholes are unequal. 5. When equals are taken from unequals the remainders are unequal.

6. Things which are double the same thing, or 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. Every whole is equal to all its parts taken together.

47. POSTULATES.

Let it be granted

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

2. That a straight line can be produced to any distance, or can be terminated at any point.

3. That the circumference of a circle can be described about any

... therefore.

48. SYMBOLS AND ABBREVIATIONS.

Post. postulate.

=

is (or are) equal to.

Zangle.

angles.

A triangle.

A triangles.
Il parallel.
parallelogram

s parallelograms.
perpendicular.

Is perpendiculars.

rt. right angle. rt. right angles.

> is (or are) greater than. <is (or are) less than.

rt. A right triangle.

rt. A right triangles.

O circle.

circles.

+ increased by.

diminished by.

X multiplied by.

divided by.

Def. definition.

Ax. axiom.

Hyp. hypothesis.

Cor. corollary.

Q. E. D. quod erat demonstran

dum.

Q. E. F. quod erat faciendum. Adj. adjacent.

Ext.-int. exterior-interior.

Alt.-int. alternate-interior.

Iden. identical.

Cons. construction.

Sup. supplementary.

Sup. adj. supplementary-adja

cent.

Ex. exercise.

Ill. illustration.

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PERPENDICULAR AND OBLIQUE LINES.

15

ON PERPENDICULAR AND OBLIQUE LINES.

PROPOSITION I. THEOREM.

49. When one straight line crosses another straight line the vertical angles are equal.

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:: Z O CA + ZOC B = LOCA + Z ACP.

Ax. 1.

Take away from each of these equals the common ≤ OC A.

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50. COROLLARY. If two straight lines cut one another, the four angles which they make at the point of intersection are

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