GEOMETRY.

INTRODUCTION.

IF, without regarding the qualities of bodies, viz: their smoothness, roughness, color, compactness, tenacity, &c., we merely consider the space which they fill—their extension in space-they become the special subject of mathematical investigation, and the science which treats of them, is called Geometry.

The extensions of bodies are called dimensions. Every body has three dimensions, viz: length, breadth, and depth. Of a wall or a house, for instance, you can form no idea, without conceiving it to extend in length, breadth, and depth; and the same is the case with every other body you can think of.

The limits or confines of bodies are called surfaces (superfices), and may be considered independently of the bodies themselves. So you may look at the front of a house, and inquire how long and how high is that house, without regarding its depth; or you may consider the length and breadth of a field, without asking how deep it goes into the ground, &c. In all such cases, you merely consider two dimensions. A surface is, therefore, defined to be an extension in length and breadth without depth.

The limits or edges of surfaces are called lines, and may again be considered independently of the surfaces themselves. You may ask, for instance, how long is the front of such a house, without regarding its height; or how far is it from Boston to Roxbury, without inquiring how broad is the road. Here, you consider evidently only one dimension; and a line, therefore, is defined to be an extension in length without breadth or depth.

The beginning and end of lines are called points. They merely mark the positions of lines, and can, therefore, of themselves, have no magnitude. To give an example: when you set out from Boston to Roxbury, you may indicate the place you start from, which you may call the point of starting. If this chances to be Marlborough Hotel, you do not ask how long, or broad, or deep that place is; it suffices for you to know the spot where you begin your journey. A point is, therefore, defined to be mere position, without either length or breadth.

Remark. A point is represented on paper or on a board, by a small dot. A line is drawn on paper with a pointed lead pencil or pen; and on the board, with a thin mark made with chalk. The extensions of surfaces are indicated by lines; and bodies are represented on paper or on the board, according to the rules of perspective.

Before we begin the study of Geometry, it is necessary, first, to acquaint ourselves with the meaning of some terms, which are frequently made use of in books treating on that science.

Definitions.

A line is called straight, when every part of it lies in the same direction, thus,

Any line in which no part is straight, is called a curve line.

A geometrical plane is a surface, in which two points being taken at pleasure, the straight line joining them lies entirely in_that_surface.* A surface in which no part is plane, is called a curved surface. Any plane surface, terminated by lines, is called a geometrical figure.

The simplest rectilinear figure, terminated by three straight lines, is called a triangle.

A geometrical figure, terminated by four straight lines, is called a quadrilateral—by 5, a pentagon—by 6, a hexagon-by 7, a heptagon-by 8, an octagon, &c.

Any geometrical figure, terminated by more than three straight lines, is (by some authors) called a polygon.†

When two straight lines meet, they form an angle; the point at which they meet is called the vertex, and the lines themselves are called the legs of the angle. When a straight line meets another, so as to make the two adjacent angles equal, the angles are called right an

* The teacher can give an illustration of this definition, by taking anywhere on a piece of pasteboard, two points and joining them by a piece of stiff wire. Then, by bending the board, the wire, which represents the line, will be off the board, and you have a curved surface; and by stretching the board, so as to make the wire fall upon it, you have a plane.

+ Legendre calls all geometrical figures polygons,

gles, and the lines are said to be perpendicular to each other.

Any angle smaller than a right angle is called acute,

and when greater than a right angle, an obtuse angle.*

Two lines which, lying in the same plane, and however far extended in both directions, never meet, are said to be parallel to each other.

When two lines, situated in the same plane, are not parallel, they are either converging or diverging. Two lines are said to be converging, if, when extended in the direc tion we consider, they grow nearer each other; and diverging, if the reverse takes place.

* Angles are measured by arcs of circles, described with any radius between their legs. Here the teacher may state, that the circle is divided into 360 equal parts, called degrees; each degree, again, into 60 equal parts, called minutes; a minute, again, subdivided into 60 equal parts, called seconds, &c.; and that the magnitude of an angle can thus be expressed in degrees, minutes, seconds, &c. of an arc of a circle, contained between its legs.

A triangle is called equilateral, when all its sides are equal.

A triangle is called isosceles, when two of its sides only are equal.

A triangle is called scalene, when none of its sides are equal.

A triangle is also called right-angled, when it contains a right angle;

and oblique-angled, when it contains no right angle.

A parallelogram is a quadrilateral whose opposite sides are parallel.

A rectangle, or oblong, is a right-angled parallelogram.