EUCLID'S ELEMENTS.

BOOK I.

DEFINITIONS.

1. A point is that which has position, but no magnitude.

3.

A line is that which has length without breadth.

The extremities of a line are points, and the intersection of two lines is a point.

4. A straight line is that which lies evenly between its extreme points.

Any portion cut off from a straight line is called a segment of it. 5. A surface (or superficies) is that which has length and breadth, but no thickness.

6. The boundaries of a surface are lines.

7. A plane surface is one in which any two points being taken, the straight line between them lies wholly in that surface.

A plane surface is frequently referred to simply as a plane.

NOTE. Euclid regards a point merely as a mark of position, and he therefore attaches to it no idea of size and shape.

Similarly he considers that the properties of a line arise only from its length and position, without reference to that minute breadth which every line must really have if actually drawn, even though the most perfect instruments are used.

The definition of a surface is to be understood in a similar way.

H.S.E.

8. A plane angle is the inclination of two lines to one another, which meet together, but are not in the same direction.

[Definition 8 is not required in Euclid's Geometry, the only angles employed by him being those formed by straight lines. See Def. 9.]

9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

The point at which the straight lines meet is called the vertex of the angle, and the straight lines themselves the arms of the angle.

NOTE. When there are several angles at one point, each is expressed by three letters, of which the letter that refers to the vertex is put between the other two. Thus the angle contained by the straight lines OA, OB is named the angle AOB or BOA; and the angle contained by OA, OC is named the angle AOC or COA. But if there is only one angle at a point, it may be expressed by a single letter, as the angle at O.

Of the two straight lines OB, OC shewn in the adjoining diagram, we recognize that OC is more inclined than OB to the straight line OA: this we express by saying that the angle AOC is greater than the angle AOB. Thus an angle must be regarded as having magnitude.

B

A

B

A

It must be carefully observed that the size of an angle in no way depends on the length of its arms, but only on their inclination to one another.

The angle AOC is the sum of the angles AOB and BOC; and AOB is the difference of the angles AOC and BOC.

[Another view of an angle is recognized in many branches of mathematics; and though not employed by Euclid, it is here given because it furnishes more clearly than any other a conception of what is meant by the magnitude of an angle.

;

Suppose that the straight line OP in the diagram is capable of revolution about the point O, like the hand of a watch, but in the opposite direction and suppose that in this way it has passed successively from the position OA to the positions occupied by OB and OC. Such a line must have undergone more turning in passing from OA to

C

B

A

OC, than in passing from OA to OB; and consequently the angle AOC is said to be greater than the angle AOB.]

Angles which lie on either side of a common

arm are called adjacent angles.

For example, when one straight line OC is drawn from a point in another straight line AB, the angles COA, COB are adjacent.

A

When two straight lines, such as AB, CD, cross one another at E, the two angles CEA, BED are said to be vertically opposite. The A two angles CEB, AED are also vertically opposite to one another.

10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

11. An obtuse angle is an angle which is greater than a right angle.

12. An acute angle is an angle which is less than a right angle.

[In the adjoining figure the straight line OB may be supposed to have arrived at its present position, from the position occupied by OA, by revolution about the point Ò in either of the two directions indicated by the arrows thus two straight lines drawn from a point may be considered as forming two angles (marked (i) and (ii) in the figure), of which the greater (ii) is said to be reflex.

If the arms OA, OB are in the same straight line, the angle formed by them on either side is called a straight angle.]

E

B

A

13. A term or boundary is the extremity of anything. 14. Any portion of a plane surface bounded by one or more lines is called a

plane figure.

The sum of the bounding lines is called the perimeter of the figure. Two figures are said to be equal in area when they enclose equal portions of a plane surface.

15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another; this point is called the centre of the circle.

16. A radius of a circle is a straight line drawn from the centre to the circumference.

17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

18. A semicircle is the figure bounded by a diameter of a circle and the part of the circumference cut off by the diameter.

19. A segment of a circle is the figure bounded by a straight line and the part of the circumference which it cuts off.

20.

Rectilineal figures are those which are bounded

by straight lines.

21. A triangle is a plane figure bounded.

by three straight lines.

Any one of the angular points of a triangle may be regarded as its vertex; and the opposite side is then called the base.

22. A quadrilateral is a plane figure bounded by four straight lines.

The straight line which joins opposite angular points in a quadrilateral is called a diagonal.

23. A polygon is a plane figure bounded by more than four straight lines.

TRIANGLES.

24. An equilateral triangle is a triangle whose three sides are equal.

25. An isosceles triangle is a triangle two of whose sides are equal.

26. A scalene triangle is a triangle which has three unequal sides.

27. 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.

28. An obtuse-angled triangle is a

triangle which has an obtuse angle.

29. 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.]