These symbols take the plural form when necessary, as in the case of Ils, 4, A, 9.

X, are used as in algebra.

The symbols +, There is no generally accepted symbol for “is congruent to," and the words are used in this book. Some teachers use or, and some use , but the sign of equality is more commonly employed, the context telling whether equality, equivalence, or congruence is to be understood. Q. E. D. is an abbreviation that has long been used in geometry for the Latin words quod erat demonstrandum, "which was to be proved." Q. E. F. stands for quod erat faciendum, "which was to be done."

* viii

PLANE GEOMETRY

INTRODUCTION

1. The Nature of Arithmetic. In arithmetic we study computation, the working with numbers. We may have a formula expressed in algebraic symbols, such as a=bh, where a may stand for the area of a rectangle, and b and h respectively for the number of units of length in the base and height; but the actual computation involved in applying such formula to a particular case is part of arithmetic.

2. The Nature of Algebra. In algebra we generalize the arithmetic, and instead of saying that the area of a rectangle with base 4 in. and height 2 in. is 4 x 2 sq. in., we express a general law by saying that a = bh. In arithmetic we may have an equality, like 2 x 16+17=49, but in algebra we make much use of equations, like 2x+17=49. Algebra, therefore, is a generalized arithmetic.

3. The Nature of Geometry. We are now about to begin another branch of mathematics, one not chiefly relating to numbers although it uses numbers, and not primarily devoted to equations although using them, but one that is concerned principally with the study of forms, such as triangles, parallelograms, and circles. Many facts that are stated in arithmetic and algebra are proved in geometry. For example, in geometry it is proved that the square on the hypotenuse of a right triangle equals the sum of the squares on the other two sides, and that the circumference of a circle equals 3.1416 times the diameter.

4. Solid. The block here represented is called a solid; it is a limited portion of space filled with matter. In geometry, however, we have nothing to do with the matter of which a

body is composed; we study simply its shape and size, as in the second figure.

That is, a physical solid can be touched and handled; a geometrio solid is the space that a physical solid is conceived to occupy. For example, a stick is a physical solid; but if we put it into wet plaster, and then remove it, the hole that is left may be thought of as a geometric solid although it is filled with air.

5. Geometric Solid. A limited portion of space is called a geometric solid.

6. Dimensions. The block represented in § 4 extends in three principal directions:

(1) From left to right, that is, from A to D;
(2) From back to front, that is, from A to B;

(3) From top to bottom, that is, from A to E.

These extensions are called the dimensions of the block, and are named in the order given, length, breadth (or width), and thickness (height, altitude, or depth). Similarly, we may say that every solid has three dimensions.

Very often a solid is of such shape that we cannot point out the length, or distinguish it from the breadth or thickness, as an irregular block of coal. In the case of a round ball, where the length, breadth, and thickness are all the same in extent, it is impossible to distinguish one dimension from the others.