means straight-lined or right-lined (Lat. rectus, right; linea, a line). In many modern books on geometry a straight line is termed a 'right line.' No. 9 gives a general definition of the three kinds of plane

angles, right angles, angles greater than a right angle, called obtuse (Lat. obtusus, blunted) angles, smaller than a right angle, called acute (Lat. acutus, sharp). These are thus defined by Euclid :

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 that which is greater than a right angle.'

12. 'An acute angle is that which is less than a right angle.'

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OF FIGURES.

13. A term or boundary is the extremity of anything.' 14. A figure is that which is enclosed by one or more boundaries.'

15. A circle is a plane figure contained by one line called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal.' These lines are called radii (Lat. radius, the spoke of a wheel, which each resembles).

of the circle.'

16. And this point is called the centre

17. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference' (equal in length to two radii).

18. A semicircle is the figure contained by a diameter, and that part of the circumference (called from its shape an arc, Lat. arcus), which it cuts off.'

19. A segment of a circle is the figure contained by a straight line (called a chord) and the

circumference which it cuts off.'

A semicircle is of course a segment of a circle. A sector, or section, of a circle is the figure contained by two radii and the arc cut off.

The circumference of a circle is nearly 34 times (3.14159) the diameter. The area of a circle may be found by multiplying the square of the radius by 34, the relation of the diameter to the circumference (πp2), e.g. supposing the radius of a circle to be 6 in. in length, its area would be (36 × 34) sq. in. Neither of these statements can be fully proved without some knowledge of mathematics, but their truth may be roughly ascertained, in the first instance by actual measurement, and in the second thus :Draw a circle of any size; draw it in two diameters cutting each other at right angles. The two diameters cutting at the centre will be divided into four radii, on each of these draw a square, the resulting figure will be equal to four times the square of the radius, and we shall see that the circle does not quite fill it up; hence we should expect that the area of the circle would be equal to rather more than three times the square of the radius.

The circumference of a circle is divided by geometers into 360 equal parts, called degrees. This division will be familiar to most of you from your knowledge of geography; the circles which we imagine to pass round the earth's surface, forming the parallels of latitude and longitude, are divided, as you know, into 360°. If you will refer a moment to the figure above you will see that the four right angles, formed by the diameters cutting each other, each intercept or cut off a quarter of the circumference of the circle, containing, as you

know, 90°. A right angle is therefore often spoken of as an angle of 90°, and angles as well as circles are measured by degrees according to the size of the arc which they intercept. An acute angle being less than a right angle may contain any number of degrees less than 90; an obtuse angle, being greater than a right angle, may contain any number of degrees between 90 and 180.

OF RECTILINEAL FIGURES.

You will readily see that three straight lines are the smallest number that will contain a figure, and also that a figure contained by straight lines will have as many angles as sides; hence these figures may be named either according to their angles or their sides. Thus Euclid speaks of trilateral, three-sided, quadrilateral, four-sided, multilateral, many-sided, and equilateral, equal-sided, figures. Again we have triangle, a three-cornered or three-angled figure, polygon, a manyangled figure (Gr. Toλùs, many, ywvía, an angle), also pentagon, a five-angled figure, hexagon, octagon, &c.

20. 'Rectilineal figures are those which are contained by straight lines.'

21. Trilateral figures, or triangles, are contained by three straight lines.'

22. Quadrilateral figures by four straight lines.'

23. Multilateral figures, or polygons, by more than four straight lines.'

A regular polygon has all its sides equal. The perimeter of any figure is the measurement round about it, i.e. the sum of its sides.

OF THREE-SIDED FIGURES.

Euclid classifies triangles (1) according to their sides, as equilateral or equal sided, isosceles (ioos, equal, σkéλoç, leg), equal legged, having two equal sides, and scalene, limping (Gr. σkaλnyós), having three unequal sides; (2) according to their angles, as right-angled, obtuse-angled, and acute-angled. It will be shown further on that every triangle contains two

acute angles; consequently a right-angled triangle has one right angle, an obtuse-angled triangle an obtuse angle, and an acute-angled triangle three acute angles.

24. An equilateral triangle is that which has three equal sides.'

25. 'An isosceles triangle is that which has two sides equal.'

26. 'A scalene triangle is that which has three unequal sides.'

27. A right-angled triangle is that which has a right angle.'

28. 'An obtuse-angled triangle is that which has an obtuse angle.'

29. An acute-angled triangle is that which has three acute angles.'

OF FOUR-SIDED FIGURES.

Four-sided figures are of two classes, those having their opposite sides parallel, called parallelograms, and irregular figures, called trapeziums (tables).

The line joining the opposite angles of any quadrilateral is called a diameter, or diagonal. Parallelograms are of four kinds, which are thus defined by Euclid :

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:

30. A square is that which has all its sides equal and all its angles right angles.'

31. An oblong, or rectangle, is that which has all its angles right angles, but not all its sides equal.'

32. A rhombus is that which has all its sides equal, but all its angles are not right angles.'

33. A rhomboid is that 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 besides these are called trapeziums.'

OF PARALLELS.

35. 'Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet.

NOTE. The parts of the circle enumerated in defs. 17, 18, 19, are not referred to afterwards in Book I., and the figures defined in Nos. 31, 32, 33, are spoken of generally as parallelograms.

THE POSTULATES.

The postulates, in the original airhuara, requests or petitions, as the word is translated in the old editions of Euclid, come next. In the first three Euclid asks:

to

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

2. That a terminated straight line may be produced to any length in a straight line.'

3. That a circle may be described from any centre at any distance from that centre.'

Postulates 4 and 5 refer to simple geometrical truths which are quite evident; postulate 6 is not so evident and will be discussed later on-it is inserted here to make the list complete. These three are in many editions placed among the axioms.

4. 'Let it be granted that all right angles are equal to one

another.'

5. That two straight lines cannot enclose a space.'