PRACTICAL GEOMETRY.

We will not commence our instructions with the hackneyed “definitions,” but give our readers full credit for the knowledge of what is a point, a right or straight line, a curved line, parallel lines—and so forth, and proceed at once to practice.

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.

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.

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.

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.

Take the radius of the circle, and with it mark off six points on the circumference. Take two of these lengths of the radius and join their extreme points A and B, which will be the base. Now take this base as a radius and describe alternately two arcs cutting each other at C. Join A, C, and B, C, and a triangle is formed, whose sides being equal is termed an equilateral triangle.

In order to ensure its being upright, erect a perpendicular at the centre, and let the two sides A, C, and B, C, meet that perpendicular where it intersects the circumferences. Or, begin the triangle at this point, and mark off two lengths of the radius, joining the extreme points as before; and do this at each side of the perpendicular; finally connecting the distant extremities of the two sides for a base.

Problem V. To construct an upright square in a given circle.

Let fall a perpendicular, I, E, from the centre to the circumference, and with that as a radius and E as a centre, cut the circumference at A, B, C, and D, and join the points. The four-sided figure called a square is thus formed.

Problem VI. On a given right line, A, B, to construct a pentagon, or five-sided figure.

Draw B, F, perpendicular and equal to the half of A, B. Produce A, F, to G, making F, G, equal to F, B. From the points A and B, with the radius B, G, describe arcs cutting each other at I. From I, with the radius I, B, describe a circle. Inscribe the successive chords A, E; E, D; D, C; C, B, which with the base A, B, completes the pentagon.

If the circle be given, and a pentagon to be inscribed in it, the following is as simple as it is practical. From the centre erect a perpendicular, which shall meet the circumference at D. At each side of this point divide the circumference into five equal parts, and connect every two of them from D to E, from E to A, and from D to C, C to B. Now connect A and B and the pentagon is formed.

Problem VII. On a given line A, B, to construct a hexagon, or six-sided figure. Take the length of the radius I, G, and lay it off from F to A, A to B, B to C, C to D, D to E, and E to F.

Problem VIII. To form an octagon, or eight-sided figure.

Refer back to Fig. 5. Draw the radius I, E, till it meets the circumference at E. Join the points E, A, and E, B. Repeat this at each of the four sides, and the octagon is formed.

Problem IX. To form a decagon, or ten-sided figure.

Refer to Fig. 6, and proceed as in the preceding problem.

Problem X. To construct a duo-decagon, or twelve-sided figure.

Refer to Fig. 7, and duplicate the chords, as already shown.

We do not present 7, 9, or 11 sided figures, because they seldom or ever come into practice. Our object being to give what is useful and not overburden the memory unnecessarily.

The learner should go over and work out each of the foregoing problems several times. In fact, until they are soundly secured in his memory, so that on any emergency he can apply them to a required practice. They are the simplest rudiments, but as practically useful as they are simple. The Architect, the builder, as well as the several trades of carpenter, joiner, carver, stone-cutter, mason, and in fact, all in any way concerned in the practice of construction will at some time or other wish to recall one of these useful problems. Therefore do we dwell on the necessity for committing them, understandingly, to memory, and likewise the advantage required in being able to draw them neatly and perfectly on paper. In order to do this with satisfaction to one’s self, it is desirable that a fine point be constantly maintained on the pencil, and that uniform nicety be preserved with the curved lines, as well as the right or straight lines. For nothing looks worse than undue thickness in the one or the other. All should be alike.

In theoretical geometry a line, whether right or curved, is but imaginary, not having any thickness whatever, and therefore no palpable existence. In practical geometry the line must be visible, but ought to be so uniformly fine as to occupy scarcely any perceptible thickness. And herein lies the greatest beauty in geometrical draughting. By strict attention to this apparently trifling matter, its advantages will show wherever minute angles occur. They will be clear and distinct, and always satisfactory.

The learner should keep his first attempts, however coarse, for they will by comparison hereafter, show the advance he has made. Nor should he be content to “let well enough alone.” There is no “well enough” in drawing. It is a progressive science, and the true artist never believes he has done his best. Go as near to perfection as you can, and do not turn aside from, or step over obstacles to reach the end you have in view. Whatever you have neglected in early study will surely haunt you through after years, and trouble you when you can least bear the annoyance.

We now conclude this primary lesson, hoping that our learners may profit by the hints we have thrown out, and will thoroughly prepare themselves for the advance in our next.

The first brick house in Iowa was built by Judge Rerer, of Burlington, in 1839.