BOOK I.

THE STRAIGHT LINE.

DEFINITIONS.

Def. 1. A point has position, but it has no magnitude. Def. 2. A line has position, and it has length, but has neither breadth nor thickness. The extremities of a line are points, and the intersection of two lines is a point.

Def. 3. A surface has position, and it has length and breadth, but not thickness. The boundaries of a surface, and the intersection of two surfaces, are lines.

Def. 4. A solid has position, and it has length, breadth and thickness.

The boundaries of a solid are surfaces.

Def. 5. A straight line is such that any part will, however placed, lie wholly on any other part, if its extremities are made to fall on that other part.

Def. 6. A plane surface, or plane, is a surface in which any two points being taken the straight line that joins them lies wholly in that surface.

Def. 7. A plane figure is a portion of a plane surface enclosed by a line or lines.

Def. 8. 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.

Def. 9. A radius of a circle is a straight line drawn from the centre to the circumference.

Def. 10. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference.

Def. II. When two straight lines are drawn from the same point, they are said to contain, or to make with each other, a plane angle. The point is called the vertex, and the straight lines are called the arms, of the angle.

An angle is a simple concept incapable of definition, properly so called, but the nature of the concept may be explained as follows, and for convenience of reference it may be reckoned among the definitions.

A

B

C

A line drawn from the vertex and turning about the vertex in the plane of the angle from the position of coincidence with one arm to that of coincidence with the other is said to turn through the angle: and the angle is greater as the quantity of turning is greater. Since the line may turn from the one position to the other in either of two ways, two angles are formed by two straight lines drawn from a point. These angles (which have a common vertex and common arms) are said to be conjugate. The greater of the two is called the major conjugate, and the smaller the minor conjugate, angle.

When the angle contained by two lines is spoken of without qualification, the minor conjugate angle is to be understood. It is seldom requisite to consider major conjugate angles before Book III.

When the arms of an angle are in the same straight line, the conjugate angles are equal, and each is then said to be a straight angle.

An angle is named by a single letter at its vertex, as A : or by a letter at the vertex placed between letters at points on each of its arms, as BAC, or CAB.

Def. 12. When three straight lines are drawn from a point, if one of them be regarded as lying between the other two, the angles which this one (the mean) makes with the other two (the extremes) are said to be adjacent angles: and the angle between the extremes, through which a line would turn in passing from one extreme through the mean to the other extreme, is the sum of the two adjacent angles.

Thus AOB, BOC are adjacent,

and AOB+ BOC=AOC,

also AOC- COB=AOB.

C

B

Def. 13. The bisector of an angle is the straight line that divides it into two equal angles.

Def. 14. When one straight line stands upon another straight line and makes the adjacent angles equal, each of the angles is called a right angle.

OBS. Hence a straight angle is equal to two right angles; or, a right angle is half a straight angle, and a straight line makes with its continuation at any point an angle of two right angles.

Def. 15. A perpendicular to a straight line is a straight line that makes a right angle with it.

Def. 16. An acute angle is that which is less than a right angle.

Def. 17. An obtuse angle is that which is greater than one right angle, but less than two right angles.

Def. 18. A reflex angle is a term sometimes used for a major conjugate angle.

Def. 19. When the sum of two angles is a right angle, each is called the complement of the other, or is said to be complementary to the other.

Def. 20. When the sum of two angles is two right angles, each is called the supplement of the other, or is said. to be supplementary to the other.

Def. 21. The opposite angles made by two straight lines that intersect are called vertically opposite angles.

Def. 22. A plane rectilineal figure is a portion of a plane surface inclosed by straight lines. When there are more than three inclosing straight lines the figure is called a polygon.

Def. 23. A polygon is said to be convex when no one of its angles is reflex.

Def. 24. A polygon is said to be regular when it is equilateral and equiangular; that is, when all its sides and angles are equal.

Def. 25. A diagonal is the straight line joining the vertices of any angles of a polygon which have not a common

arm.

Def. 26. The perimeter of a rectilineal figure is the sum of its sides.

Def. 27. The area of a figure is the space inclosed by its boundary.

Def. 28. A triangle is a figure contained by three straight lines.

Def. 29. A quadrilateral is a polygon of four sides, a pentagon one of five sides, a hexagon one of six sides, and so

on.

GEOMETRICAL Axioms.

1. Magnitudes that can be made to coincide are equal.

2. Two straight lines that have two points in common lie wholly in the same straight line.

3. A finite straight line has one and only one point of bisection.

4. An angle has one and only one bisector.

POSTULATES.

Let it be granted that

I. A straight line may be drawn from any one point to any other point.

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

3. A circle may be described from any centre, with a radius equal to any finite straight line.

It will be seen that these postulates amount to a request to use the straight edge of a ruler, and a pair of compasses; the latter being such that a distance can be carried by them from one part of the paper to another.

It may be useful to have a list of the derivations of some of the common terms used in geometry:

Axiom. ağiwua, a statement deemed true.

Theorem. θεώρημα.

Hypothesis. óleois, a supposition, a foundation, from vπò, τí☺ŋμi.

Identity. Idem, identidem, the same thing.

Geometry. n, μeтpéw, to measure land.

Diameter. did μerpéw, to measure across.