There are some persons who think that with a drawing-board and square, they can, without fail, make all sorts of horizontal, perpendicular, or parallel lines, and that therefore any geometrical rules for such purpose are to them unnecessary. But, suppose the drawing-board, or the square is absent, or that neither can be had. In such an emergency the want of the following items of knowledge would be severely felt, and, therefore, the acquirement and retention of them is something desirable, and even highly necessary. PROBLEM I. _To erect a perpendicular on a given right line._ [Illustration: _Fig. 1_] A, B, is the given right line. From the point C, with a radius longer than the perpendicular distance describe the arc, or part of a circle, D, D. And from the points of intersection with the right line A, B, describe arcs cutting each other at C and E. Join C and E, and the perpendicular is obtained on either side of the right line A, B. PROBLEM II. _To erect a perpendicular at the middle of a right line._ From the extreme points of the right line A, B, with radii less than the length of the line describe two arcs intersecting each other at C and D, and through the points of their intersection draw the line, which will be perpendicular to the given right line at the middle. [Illustration: _Fig. 2_] In this way, too, may any line be divided into too equal parts with facility and exactness. PROBLEM III. _To erect a perpendicular at or near the end of a given right line._ [Illustration: _Fig. 3_] Take any point, D, on the given right line A, B, as a centre, and to the required point C, as a radius, and describe an arc C, E, F. Take a portion of this arc, say E, and make from C, E, equal to E, F. Join F and C. Now with E, C, for a radius, describe the arc G, E, H, and make from E to H equal to from E to G. Then through H from C draw the perpendicular required. There are other methods of accomplishing this, but we will not introduce them here, as the one now given is sufficient. We will now proceed to the formation of geometrical figures which enclose space. That which is bounded by one line is called a _circle_; and a right line dividing it into two equal parts is called its _diameter_; from the centre of which to either end is called the _radius_: and the boundary line is termed the _circumference_ from the Latin words _circum_, around, and _fero_ to carry. That is: a line carried around. Thus we see an area or space is enclosed by one line. An area may be enclosed by two lines; but one, or both of them, must be curved; as two right lines cannot enclose a space. But three can; and the figure is called a _triangle_. PROBLEM IV. _In a given circle to construct a Triangle._ [Illustration: _Fig. 4_]