QUADRILATERALS.

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

[It may 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.]

31. An oblong is a four-sided figure which has all its angles right angles, but not all its sides equal.

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

33. A rhomboid is a four-sided figure which has its opposite sides equal to one another, but all its sides are not equal nor its angles right angles.

34. All other four-sided figures are called trapeziums.

It is usual now to restrict the term trapezium to a quadrilateral which has two of its sides parallel. [See Def. 35.]

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

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

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

Let it be granted,

THE POSTULATES.

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

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.

NOTES ON THE POSTULATES.

1. In order to draw the diagrams required in Euclid's Geometry certain instruments are necessary. These are

(i) A ruler with which to draw straight lines.

(ii) A pair of compasses with which to draw circles.

In the Postulates, or requests, Euclid claims the use of these instruments, and assumes that they suffice for the purposes mentioned above.

2. 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 are not to be employed for transferring distances from one part of a diagram to another.

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

To produce a straight line means to prolong or lengthen it.

The expression to describe is used in Geometry in the sense of to draw.

ON THE AXIOMS.

The science of Geometry is based upon certain simple statements, the truth of which is so evident that they are accepted without proof.

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

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.

10. Two straight lines cannot enclose a space.

11. All right angles are equal.

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.

That is to say, if the two straight A lines AB and CD are met by the straight line EH at F and G, in such a way that the angles BFG, DGF are together less than two right angles, it is asserted that AB and CĎ will meet if continually produced in the direction of B and D.

B

NOTES ON THE AXIOMS.

1. 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.

2. Euclid's Axioms may be classified as general and geometrical. General Axioms apply to magnitudes of all kinds. Geometrical Axioms refer specially to geometrical magnitudes, as lines, angles, and figures.

3. Axiom 8 is Euclid's test of the equality of two geometrical magnitudes. It implies that any line, angle, or figure, may 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, and it states that two such magnitudes are equal when one can be exactly placed over the other without overlapping.

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

4. 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.

INTRODUCTORY.

1. Little is known of Euclid beyond the fact that he lived about three centuries before Christ (325-285) at Alexandria, where he became famous as a writer and teacher of Mathematics.

Among the works ascribed to him, the best known and most important is The Elements, written in Greek, and consisting of Thirteen Books. Of these it is now usual to read Books I.-IV. and VI. (which deal with Plane Geometry), together with parts of Books XI. and XII. (on the Geometry of Solids). The remaining Books deal with subjects which belong to the theory of Arithmetic.

2. 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.

3. The subject is divided into a number of separate discussions, called propositions.

Propositions are of two kinds, Problems and Theorems.

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

A Theorem proposes to prove the truth of some geometrical

statement.

4. A Proposition consists of the following parts :

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

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

(ii) The Particular Enunciation repeats in special terms the statement already made, and refers it to a diagram, which enables the reader to follow the reasoning more easily.

(iii) The Construction then directs the drawing of such straight lines and circles as may be required to effect the purpose of a problem, or to prove the truth of a theorem.

(iv) The Proof shews that the object proposed in a problem has been accomplished, or that the property stated in a theorem is true.