importance of graphical methods is the work of page 281 on areas. Attention is invited to Theorem 2, Book III, a simple proof of which is given not involving the use of the theorem of limits. In the other theorems usually proved by the method of limits the meaning of the theorem has been brought out by illustration and the student convinced of its truth without any attempt at proof. This is in accord with a widespread feeling that the method of limits is unprofitable for the majority of American students when they first study the plane geometry. The plan also avoids the common error of designating as a proof a line of thought which is not a proof.

One hundred and eighty additional exercises, grouped according to the books on which they depend, will be found at the end of the text.

PLANE GEOMETRY

THE ORIGIN OF GEOMETRY

The elementary geometry of to-day is mainly due to the genius of the Greeks. The foundation for their work, however, was obtained from the Babylonians and the Egyptians.

The Babylonian knowledge of geometry was developed, in part at least, through the necessity of constructing figures having religious significance. These figures involved triangles, quadrilaterals, circles, and inscribed regular hexagons. Their value 3 for π was much less accurate than 3.1604, which was used by the Egyptians probably as early as 3000 B.C.

The Egyptians were great builders. We read that Menes, their first king, who built the temple of Ptah at Memphis, also constructed a great reservoir and even changed the course of the Nile. The ruins of many temples, the Sphinx, and the great pyramids of Gizeh all attest considerable insight into geometric relations. The geometry they developed was applied to the calculation of the contents of granaries, to the laying out of right angles, to obtaining north-and-south lines for their temples and pyramids, and to carrying out land surveys. Herodotus, the Greek historian, who traveled in Egypt, says that the overflow of the Nile necessitated an annual resurvey of the land along its banks for the double purpose of determining ownership and of justly levying taxes. The word geometry itself is of Greek origin and signifies. earth measurement."

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Our information concerning the Egyptians goes back with certainty to 1700 B. C. and with great probability to 3000 B. C. We know that two thousand, perhaps three thousand, years before our era they had formulas a few of which were incorrect for the areas of triangles, rectangles, parallelograms, and trapezoids, and fairly accurate knowledge of the area of a circle. They knew also that a triangle is a right triangle if its sides are 3, 4, and 5 units respectively. Such geometry as they had grew out of their practical needs and was embodied in working rules. It was very useful, but vastly inferior to the scientific geometry afterwards developed by the Greeks. In judging the intellectual attainments of any people, however, we must remember that first advances in any science are the most difficult. Other peoples had the same practical needs as the Egyptians but developed no geometry to meet them. The Aztecs of Mexico, for example, were skillful artificers in silver and gold, and they transported huge blocks of stone from distant quarries to build their splendid temples, but their imagination never led them to construct a vehicle on wheels, one of the simplest applications of a circle, but undoubtedly the most useful, which has ever been made.