Any symbol representing a noun is converted into the plural by affixing the letters; thus means angles. It is not necessary to learn any of these symbols until they are introduced in the text. viii GEOMETRY INTRODUCTION IN arithmetic and algebra, frequent reference is made to the rectangle, the square, the triangle, and the circle. These are geometrical figures, and in geometry a careful study of them and of many others is made. Geometrical figures are used constantly in architecture. Our playgrounds are often laid out in geometrical forms. Familiarity with such figures and their properties, and ability to construct and measure them, is both interesting and worth while. It is interesting also to know how man has developed his knowledge of such figures and his skill in using them. HISTORY OF GEOMETRY Geometry as it is now studied has been handed down to us from the Greeks. The word "geometry" is derived from two Greek words meaning the earth and to measure; this fact is evidence that the Greeks believed that geometry was intimately associated with or else had been developed out of the practical business of measuring the earth, surveying. The Greeks received their start in geometry from the Egyptians. Thales of Miletus (630-550 B.C.) is given special credit for transplanting a knowledge of Egyptian geometry to Greece. Did the Egyptians originate geometry? Whether they did or not, there is evidence that they had some knowledge of practical geometry. Their pyramids and other marvelous structures point to this fact. Also, there is in the British Museum a papyrus written about 1700 B.C. by an Egyptian, commonly called Ahmes, which contains among other interesting mathematical records some formulæ for measuring geometrical figures. This papyrus is a copy of another written before the time of Ahmes. Herodotus, a Greek traveler and historian, is said to be responsible for the story that the Egyptians developed these rules of mensuration because of the necessity of frequently surveying the lands which were inundated by the floods of the Nile. The Egyptians must have obtained their formulæ by experiment or by observation. Some of the formulæ were incorrect and their formula for measuring the area of a circle was less accurate than that developed later by the Greeks. The Greeks became interested in geometry for its own sake as well as for its usefulness. In the three hundred years following the time of Thales, geometry grew into a great science in their schools, far exceeding the geometry of the Egyptians in the number and interest of the facts discovered, and in the accuracy and usefulness of the results. Pythagoras and Plato were the leaders of two groups of students which were responsible for much of the advance made in the subject. Hippocrates (about 420 B.c.) made an attempt to prepare a text on geometry, but it remained for Euclid to write what became the standard text. Euclid lived between 330 and 275 B.C. He was one of the first and greatest mathematicians who taught at the University of Alexandria. As a teacher he felt the need of a text by which to lead beginners through the known facts of elementary geometry. He therefore gathered together and systematized these facts in a book known as the Elements. Euclid's Elements has stood as the model for all subsequent texts on the subject. During the two thousand years since the time of Euclid, geometry has been studied by all civilized peoples and has been enriched from time to time by their mathematicians. This history is so long and the details are so technical that it is unwise to attempt to give more of it at this time. |