GEOMETRY AMONG THE ANCIENTS

The oldest traces of geometry are found among the Egyptians and Babylonians. That a fund of mathematical facts was known to these people at a very early date is evidenced in the construction of the great pyramids, temples, and mausoleums, which could not have been erected without a knowledge of geometric principles. The subject was of special importance to the Egyptians because of the frequent land surveys necessitated by the overflow of the Nile. The inundation of the Nile near the pyramids of Gizeh, is shown in this illustration.

The oldest mathematical work known to exist was written by an Egyptian named Ahmes, who lived about 1700 B.C. This manuscript, which is now in the British Museum, treats of many kinds of algebraic and geometric problems, and mentions still older treatises on mathematics.

As early as the seventh century B.C. the geometry of the Egyptians became known to the Greeks. About 300 B.C., their great teacher of mathematics, Euclid, brought forth a masterpiece known as Euclid's Elements, which, until recent years, was almost the universal text book on geometry.

THE USES OF GEOMETRY

In contact with the material world, a person more or less unknowingly gains a great fund of geometric facts. The study of geometry is to call attention to these facts, to lead one to observe other similar facts, to relate and systematize them, and

to use them in solving various interesting as well as practical problems.

Geometry gives assistance in many of the sciences which determine the advance of material civilization. The architect, in planning a building, utilizes geometric principles both to

make the structure secure and to render it beautiful. The illustration of Lincoln Cathedral on the preceding page is an excellent example of this. The constructing engineer is able to design the framework of buildings and bridges capable of withstanding tremendous stress by the use of triangular bracing, as shown in the illustration on page 7. The sailor shapes his course and avoids dangerous localities on a voyage through his knowledge of the laws of geometry. The astronomer and the surveyor base their measurements and calculations upon geometric facts. The machinist relies upon the same laws in laying out his work. The carpenter applies geometry every time he uses his "square." The mechanic, the draftsman, and the electrician make frequent use of the facts established in geometry.

A knowledge of geometry brings pleasure and increased capacity for enjoyment by developing an appreciation for the beauty and utility of architecture, of art, and of engineering constructions. It enables the landscape gardener to develop plans for magnificent parks and public gardens. It also forms the basis of all civic improvement plans for making cities more sanitary and more attractive.

In mature life, persons encounter problems which cannot be solved by rule and are mastered only by resourcefulness and experience. The propositions and exercises of geometry develop self-reliance and a power of initiative because of the opportunities which they present for original thinking and investigation.

Finally, the student receives in geometry the only training in formal logic which is obtained in the secondary school. He gains a knowledge of pure argumentation and of sound reasoning and learns to appreciate the force and value of concise statement in the exact use of language.

1. Drawing a Straight Line. Place a ruler or straightedge on a piece of paper and draw a pencil along the edge so that it makes a continuous mark.

The line AB is a straight line.

A line may also be designated by a single letter. The first line is read, "the line a." The second line

is read, "the line l."

a

In drawing lines, use a hard pencil sharpened to a fine point. A straightedge may be made by folding a piece of paper and creasing the edge.

2. Point. The place where two lines meet or intersect each other is a point. The ends of a line are points. The place where a line is separated into parts is a point.

The lines HK and NK meet at the point K. The lines DF and GE intersect at the point O. The ends A and C of the line AC are points. The line AC is separated into the two parts AB and BC by the point B.

A point is represented by a dot. To aid in speaking about a point, a letter, as P, may be placed beside the dot.

The point is then read, "the point P."

•P

3. Angle. Two straight lines which meet at a point form an angle. The two lines are called the sides of the angle. The point of meeting is called the vertex of the angle.

This angle is read, "the angle HKE,"

"the angle EKH," or "the angle K." When

all three letters are read, the letter at the vertex is read between the other two.

The lines KH and KE are the sides of the angle and the point K is the vertex of the angle.

B

A

An angle may also be designated by a letter placed within the angle near the vertex. In this figure, the angle BDA may be read, "the angle y," and the angle ADE may be read, "the D angle z." The angle BDE may also be read, "the angle D."

E

4. Protractor. An instrument for determining the number of degrees in an angle, and also for drawing an angle of a given number of degrees is called a protractor.

5. Measuring an Angle. To measure an angle, place the point B of the protractor on the vertex of the angle. Make one edge of the protractor, as BD, coincide with one side of the angle, as BA. The position of the other side BC indicates the number of degrees in the angle. The angle ABC is an angle of 50°.