i. A point is that which has position but not dimensions.
A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line; hence it has no dimensions—that is, it has neither length, breadth, nor thickness.
ii. A line is length without breadth.
A line is space of one dimension. If it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness.
iii. The intersections of lines and their extremities are points.
iv. A line which lies evenly between its extreme
points is called a straight or right line, such as AB.
If a point move without changing its direction it will describe a right line. The direction in
which a point moves in called its “sense.” If the moving point continually changes its direction
it will describe a curve; hence it follows that only one right line can be drawn between two
points. The following Illustration is due to Professor Henrici:—“If we suspend a weight by a
string, the string becomes stretched, and we say it is straight, by which we mean to express
that it has assumed a peculiar definite shape. If we mentally abstract from this string all
thickness, we obtain the notion of the simplest of all lines, which we call a straight line.”
v. A surface is that which has length and breadth.
A surface is space of two dimensions. It has no thickness, for if it had any, however small,
it would be space of three dimensions.
vi. When a surface is such that the right line joining any two arbitrary
points in it lies wholly in the surface, it is called a plane.
A plane is perfectly flat and even, like the surface of still water, or of a smooth floor.—
Newcomb.