22.

A right-angled triangle is a triangle

which has a right angle.

The side opposite to the right angle in a right-angled triangle is called the hypotenuse.

23. An obtuse-angled triangle is a triangle which has an obtuse angle.

24. An acute-angled triangle is a triangle which has three acute angles.

[It will be seen hereafter (Book I. Proposition 17) that every triangle must have at least two acute angles.]

25. Parallel straight lines are such as, being in the same plane, do not meet, however far they are produced in either direction.

26. A Parallelogram is a four-sided figure which has its opposite sides parallel.

27. A rectangle is a parallelogram which has one of its angles a right angle.

28. A square is a four-sided figure which has all its sides equal and all its angles right angles.

[It may easily be shewn that if a quadrilateral has all its sides equal and one angle a right angle, then all its angles will be right angles.]

29. A rhombus is a four-sided figure which has all its sides equal, but its angles are not right angles.

30. A trapezium is a four-sided figure which has two of its sides parallel.

ON THE POSTULATES.

In order to effect the constructions necessary to the study of geometry, it must be supposed that certain instruments are available; but it has always been held that such instruments should be as few in number, and as simple in character as possible.

For the purposes of the first Six Books a straight ruler and a pair of compasses are all that are needed; and in the following Postulates, or requests, Euclid demands the use of such instruments, and assumes that they suffice, theoretically as well as practically, to carry out the processes mentioned below.

POSTULATES.

Let it be granted,

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

When we draw a straight line from the point A to the point B, we are said to join AB.

2. That a finite, that is to say, a terminated straight line may be produced to any length in that straight line.

3. That a circle may be described from any centre, at any distance from that centre, that is, with a radius equal to any finite straight line drawn from the centre.

It is important to notice that the Postulates include no means of direct measurement: hence the straight ruler is not supposed to be graduated; and the compasses, in accordance with Euclid's use, are not to be employed for transferring distances from one part of a figure to another.

ON THE AXIOMS.

The science of Geometry is based upon certain simple statements, the truth of which is assumed at the outset to be selfevident.

These self-evident truths, called by Euclid Common Notions, are now known as the Axioms.

The necessary characteristics of an Axiom are

(i) That it should be self-evident; that is, that its truth should be immediately accepted without proof.

(ii) That it should be fundamental; that is, that its truth should not be derivable from any other truth more simple than itself.

(iii) That it should supply a basis for the establishment of further truths.

These characteristics may be summed up in the following definition.

DEFINITION. An Axiom is a self-evident truth, which neither requires nor is capable of proof, but which serves as a foundation for future reasoning.

Axioms are of two kinds, general and geometrical.

General Axioms apply to magnitudes of all kinds. Geometrical Axioms refer exclusively to geometrical magnitudes, such as have been already indicated in the definitions.

GENERAL AXIOMS.

1. Things which are equal to the same thing are equal to one another.

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, the greater sum being that which includes the greater of the unequals.

5. If equals be taken from unequals, the remainders are unequal, the greater remainder being that which is left. from the greater of the unequals.

6. Things which are double of the same thing, or of equal things, are equal to one another.

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

9.* The whole is greater than its part.

* To preserve the classification of general and geometrical axioms, we have placed Euclid's ninth axiom before the eighth.

GEOMETRICAL AXIOMS.

8. Magnitudes which can be made to coincide with one another, are equal.

This axiom affords the ultimate test of the equality of two geometrical magnitudes. It implies that any line, angle, or figure, may be supposed to be taken up from its position, and without change in size or form, laid down upon a second line, angle, or figure, for the purpose of comparison.

This process is called superposition, and the first magnitude is said to be applied to the other.

10. Two straight lines cannot enclose a space.

11. All right angles are equal.

[The statement that all right angles are equal, admits of proof, and is therefore perhaps out of place as an Axiom.]

12. If a straight line meet two straight lines so as to make the interior angles on one side of it together less than two right angles, these straight lines will meet if continually produced on the side on which are the angles which are together less than two right angles.

[Axiom 12 has been objected to on the double ground that it cannot be considered self-evident, and that its truth may be deduced from simpler principles. It is employed for the first time in the 29th Proposition of Book I., where a short discussion of the difficulty will be found.

The converse of this Axiom is proved in Book I. Proposition 17.]

INTRODUCTORY.

Plane Geometry deals with the properties of all lines and figures that may be drawn upon a plane surface.

Euclid in his first Six Books confines himself to the properties of straight lines, rectilineal figures, and circles.

The Definitions indicate the subject-matter of these books: the Postulates and Axioms lay down the fundamental principles which regulate all investigation and argument relating to this subject-matter.

Euclid's method of exposition divides the subject into a number of separate discussions, called propositions; each proposition, though in one sense complete in itself, is derived from results previously obtained, and itself leads up to subsequent propositions.

Propositions are of two kinds, Problems and Theorems.

A Problem proposes to effect some geometrical construction, such as to draw some particular line, or to construct some required figure.

A Theorem proposes to demonstrate some geometrical truth.

A Proposition consists of the following parts:

The General Enunciation, the Particular Enunciation, the Construction, and the Demonstration or Proof.

(i) The General Enunciation is a preliminary statement, describing in general terms the purpose of the proposition.

In a problem the Enunciation states the construction which it is proposed to effect: it therefore names first the Data, or things given, secondly the Quæsita, or things required.

In a theorem the Enunciation states the property which it is proposed to demonstrate: it names first, the Hypothesis, or the conditions assumed; secondly, the Conclusion, or the assertion to be proved.