SECTION II.

FUNDAMENTAL DEFINITIONS.

17. DEF.

STRAIGHT LINE, PLANE, ANGLE.

A Straight Line.

Straight lines are such that when any straight line is placed so that two of its points coincide with two points of another straight line, the two lines cannot but coincide throughout their entire length.

This definition does not tell us how to draw a straight line. It gives us a test which we may apply to a line to examine whether it is straight or not.

Thus, when a carpenter is making a 'straight-edge' (such as the intersection of two of the surfaces of a long piece of wood), in order to test its straightness he applies the edge carefully to another straightedge and examines whether the two edges coincide in all possible positions.

The student should as far as possible practically test the truth of the definition. It should be noticed that lines which are not straight, may, if they are exactly equal, be applied to one another so as to coincide throughout their entire length; but unless they are straight, it will always be possible that they should have two points common, without coinciding throughout their entire length.

For example, two equal circles can be applied the one to the other so as to coincide; the circles may have two points common without coinciding. [See fig. p. 12.]

18. DEF. A finite straight line is a portion of a straight line bounded by its extreme points.

In the figure A, B indicate the extreme points bounding a picture of a finite straight line called 'AB.'

19. The size of a finite straight line is called its length.

When we are given a finite straight line, we are also given the unlimited straight line of which the finite straight line is a portion. That is, a finite straight line can be prolonged on either hand in one way only.

If a straight line AB could be prolonged in two ways as ABC and ABD we should have two straight lines ABC, ABD having two points, A and B, common which do not coincide throughout their entire length. 20. A straight line is understood to be unlimited either

way.

A finite straight line is a limited portion of an unlimited straight line.

Equal finite Straight Lines.

21. Finite straight lines are said to be equal when they can be made to coincide with one another.

Thus let AB, CD be two finite straight lines, and let CD be applied to AB so that one of its extremities C is placed upon A and the point D upon the line of which AB is a finite portion.

Then if the length of CD is equal to that of AB, the point D must come upon the point B; and if the length of CD is less than that of AB, the point D will be between A and B ;

and if the length of CD is greater than that of AB the point D will be beyond B.

22. It follows from the definition of Art. 17 that two straight lines cannot have more than one point common.

For if they have two points common they must coincide throughout their entire length. They are in fact one and the same straight line. The words of Euclid are Two Straight lines cannot enclose a surface.

A Plane.

23. DEF. A Plane surface is one in which any two points being taken, the straight line passing through them is wholly in that surface.

This definition does not tell us how to make a plane. It gives us a test which we may apply to a given surface to examine whether it is plane or no.

Thus, when a mason is laying a pavement, he uses a 'straight-edge,' which he puts down in various places on the pavement and examines whether it always touches the pavement at every point of its length; if it does not always so touch the pavement, its surface is not a plane.

A carpenter when 'planing' a plank, forces a finite straight line (namely the edge of the knife of his plane) along the upper surface of the plank, and all the wood that is above this line is cut away, so that the surface remaining touches this straight line at every point.

The student should practically test the truth of the definition. It should be noticed that a surface which is not a plane may have some pairs of points in it such that the straight line joining them lies wholly in that surface. For example a cylinder; a cone.

A plane is considered as unlimited in all directions; and in the definition the straight line joining the two points may be taken as the unlimited straight line passing through those points.

*NOTE. A straight line may be said to be traced out by a moving point which moves always in some given direction.

A plane surface may be said to be traced out by a straight line which moves always in some given direction.

Space is traced out by a plane which moves always in some given direction.
Thus a point when moving in a line has one degree of freedom.

A point when moving in a plane has two degrees of freedom; one along the line describing the plane, and when fixed on the line it has the freedom of the line itself.

A point when moving in space has three degrees of freedom.

This is what is meant when it is said that A line has one dimension'; a surface has two dimensions; and a solid has three dimensions.

Plane Geometry.

24. When the lines forming a figure are all in one and the same plane, the figure is called a plane figure.

25. Plane Geometry treats of plane figures only.

Spherical Geometry treats of figures drawn on the surfaces of spheres.

Solid Geometry treats of figures not all in the same plane.

That portion of Euclid which is explained here, treats only of figures in one plane.

Hence in what follows a figure is to be always understood to mean a plane figure.

Area.

26. DEF. A closed figure is that which completely encloses a portion of surface.

27. DEF. The size of the portion of a plane surface enclosed by a figure is called the area of the figure.

Thus the area of a figure is the amount of surface it contains. Two figures are said to be equal when the lines of one of them can be made to coincide exactly with those of the other.

Two figures which are not equal in other respects may have equal

Thus in the above figure, as we shall prove later on, the figure ABCD and the figure EFGH, have equal areas.

A figure may sometimes be divided into small parts, and the parts made to coincide with parts of another figure, and thus their areas may be compared. The area of one figure is double that of another when the second, if applied twice to the first, then has completely covered the whole of its surface and has not been applied to any part of the first more than once.

28. A figure consisting only of finite straight lines is called a rectilineal figure.

Nearly all the figures in this book are rectilineal figures, except those in which circles are represented.

29. When a point is within a closed figure any straight line drawn through that point must have two at least of its points on the enclosing figure.

Thus in each of the above figures, A is a point inside a closed figure and the straight line BAC through A cuts each closed figure in points indicated by B and C.

If two closed figures intersect they must intersect in two points at the least.

30.

When two lines have one point common the lines

are said to cut each other at that point.

The point at which they cut is also called their point of intersection.