SOLID GEOMETRY.

INTRODUCTION

528. Solid geometry may be thought of as an extension of plane geometry, with which it has much in common. The figures of plane geometry lie in a plane; while in solid geometry we consider the properties of figures that do not lie entirely in one plane. Such figures are sometimes called three

dimensional figures.

529. Space Ideas. We gain ideas concerning space and objects in space through the senses of touch, sight, and hearing. Mainly through touch and sight, we determine the size and shape of an object.

530. Difficulties. Most beginners in the study of solid geometry have difficulty in visualizing, or forming a mental image, of a three-dimensional figure from a worded description or from a drawing in a plane.

Even the same picture may present to the minds of two persons quite different images. For instance, in the figure, one person may see six blocks while another will see seven.

A little practice, however, in visualizing the blocks enables one at will to see six or seven blocks.

Frequently a student does not understand a proposition or a proof because he does not visualize the figure properly. Lines may appear to extend in a different direction than was intended, or a plane may seem to lie in front of the page when it was intended that it should appear back of the page.

Whatever device may be necessary for a clear image should be used; but care should be taken not to become too dependent upon models. A very important part of the training from the study of solid geometry is that it trains in visualizing threedimensional figures from a description or a drawing.

531. Devices to Assist in Imaging. Cardboard models may be formed as described in § 753.

Planes can be made to intersect by cutting two pieces of cardboard half in two and fitting together along the cuts. Cardboard can be used in connection with sharp pointed wires, such as hatpins. § 568.

Hatpins or sharp wires held together by small corks can be used to form open models. § 571.

Various figures can be formed with the hatpins and a piece of soft board.

532. Representation of Solid Geometry Figures. In solid geometry, the figures are usually represented on paper or on the blackboard, that is, in a plane. It is well to have some idea of how to make the figures appear solid, and to "stand out" as desired. While it is not thought best to enter into a discussion of perspective, a few general suggestions that are of assistance will be given.

The figures may be supposed to be viewed from a point a little above, and either directly in front or a little to the right or left. Parts of the figure that are rectangles, in general, appear as parallelograms in the drawing. §§ 663-5, 671, 678. Polygons appear as polygons, but reduced in one direc

tion. §§ 660, 749. Parts that are circles appear as ellipses, which become narrower as the eye of the observer is nearer to the plane in which the circle lies. §§ 689, 692, 864.

Planes are supposed to be opaque, and all lines covered by them should be left out, or, if it is desired to insert them, they should be dashed lines.

Two lines that intersect in the drawing do not necessarily intersect in the solid figure. Lines are not usually of the same length as they are drawn. The size of an angle may be quite different from the size in the drawing.

In the accompanying drawings representing cubes, the one at the left appears to have the face ABCD in front and can hardly be imaged other

wise. While the one at the right can easily be seen as if viewed from a point a little above and to the right, and so having the face ABCD in front; or it may be visualized as viewed from a point a little below and to the left, and so having the face EFGH in front.

533. The Use of Plane Geometry in Solid Geometry. All of the plane geometry is at the disposal of the student in solid geometry; but, in applying the facts of plane geometry, great care must be taken not to use them except where they will hold.

Facts of plane geometry that hold without reference to any plane are the axioms and most definitions, theorems stating that figures are congruent, and all the theorems where it is evident that it does not matter if the figures are not in the same plane.

Theorems of plane geometry that hold because the nature of the figure requires that it lie in a plane are those concern

ing a triangle, two parallel lines, a parallelogram, a circle, and all other theorems where the conditions stated determine a plane in which the parts must lie.

In all other cases, the theorems of plane geometry can be applied only after all the parts concerned are shown to lie in one plane. Under these will fall many theorems concerning parallels, perpendiculars, circles, and polygons.

EXERCISES

1. Draw the figure of § 603, omitting the lines DE, EF, and DF. Can you visualize the figure as otherwise than in one plane?

2. Draw the figure of § 572, making the side lines of plane P full. Compare with the figure of the text.

3. Draw the figure of § 628, making all lines full and of the same thickness. Compare with the figure of the text.

4. Draw the figure of § 671, making all lines of the same thickness. Does your figure appear as if viewed from below? If not, make such lines heavy as will make it appear so.

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5. Draw a figure like the one at the right in § 722, but change the lines so that it will appear as if viewed from below.

6. Draw a figure of a sphere as in § 787, but viewed from a point on a level with the center of the sphere.