But it is clear that the geometry of the Egyptians was almost entirely practical and did not go beyond the requirements of the land-surveyor, farmer or merchant. They did indeed know, as far back as 2000 B.C., that in a triangle which has its sides proportional to 3, 4, 5 the angle contained by the two smaller sides is a right angle, and they used such a triangle as a practical means of drawing right angles. They had formulae, more or less inaccurate, for certain measurements, e.g. for the areas of certain triangles, parallel-trapezia, and circles. They had, further, in their construction of pyramids, to use the notion of similar right-angled triangles; they even had a name, _se-qet_, for the ratio of the half of the side of the base to the height, that is, for what we should call the _co-tangent_ of the angle of slope. But not a single general theorem in geometry can be traced to the Egyptians. Their knowledge that the triangle (3, 4, 5) is right angled is far from implying any knowledge of the general proposition (Eucl. I., 47) known by the name of Pythagoras. The science of geometry, in fact, remained to be discovered; and this required the genius for pure speculation which the Greeks possessed in the largest measure among all the nations of the world. Thales, who had travelled in Egypt and there learnt what the priests could teach him on the subject, introduced geometry into Greece. Almost the whole of Greek science and philosophy begins with Thales. His date was about 624-547 B.C. First of the Ionian philosophers, and declared one of the Seven Wise Men in 582-581, he shone in all fields, as astronomer, mathematician, engineer, statesman and man of business. In astronomy he predicted the solar eclipse of 28 May, 585, discovered the inequality of the four astronomical seasons, and counselled the use of the Little Bear instead of the Great Bear as a means of finding the pole. In geometry the following theorems are attributed to him--and their character shows how the Greeks had to begin at the very beginning of the theory--(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground (this implies the use of similar triangles