Many junior mathematics students would be forgiven for believing that the Greek mathematician Pythagoras (c. 580 B. C) invented the concept of the right-angled triangle, such is the delirium that surrounds his theorem concerning squares in year nine. It is interesting to note however that the right-angled triangle has been in use as an aid to construction and considered as a mathematical curiosity since as far back as 2000BC. This paper will consider the development of number and geometry in the societies of the ancient world.

We shall examine the part it has played in the Indian, Mesopotamian, Egyptian and Greek societies, with a little more information about the Pythagorean School and of course Pythagoras himself. As the subsequent text refers to Pythagoras’ Theorem and Pythagorean Triples, they are explained at this stage. Pythagoras’ Theorem “In a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides” In other words, if the hypotenuse has length z and the other two sides x and y respectively, then: Pythagorean Triples

These are groups of three numbers, which satisfy the equation of Pythagoras’ Theorem. For example: {3,4,5} {5,12,13} {7,24,25} {9,40,41} {11,60,61} Analysis Man’s interest in geometrical patterns dates back to early prehistoric times. Examples have been found of plaited rushes to form rudimentary textile art. Additionally these arrangements have developed into clothing, tents and rugs. However, geometrical patterns have not been restricted solely to textiles. They have also formed a noticeable part of ancient architecture. Examples have been found on Mexican monuments and Peruvian architectural remains.

This use of geometry to beautify their surroundings suggests that the ancients had the beginnings of later scientific geometry. The first being the ability to abstract, in other words being able to identify and duplicate the simple geometric shapes occurring naturally. The second, the ability to take a three-dimensional figure and represent it two dimensionally. Essentially these people were showing an understanding of the first principles of mapping. As man’s interest in and understanding of geometry evolved, it became a tool for the solution of practical problems. For example, building a house on level ground or measuring out square fields.

Various schemes were devised that allowed man to make these things happen Babylonian Mathematics In “A History of Mathematics”, Boyer emphatically states that: “The Mesopotamian civilisations of antiquity often are referred to as Babylonian, although such a designation is not strictly correct. The city of Babylon was not at first, nor was it always in later periods, the centre of the culture associated with the two rivers [Tigris and Euphrates], but convention has sanctioned the informal name “Babylonian” for the region during the interval from about 2000BC to roughly 600BC”

Before discussing Babylonian geometry, we must first take a look at the history of Babylonian mathematics generally. As far back as 3000BC, the Sumerians were building homes and temples in the area and decorating them with geometrical mosaics. They had already developed a system of cuneiform writing a millennium before and there is evidence that writing had been in existence in the area as far back as 5000BC. A significant event in the history of the area, however, comes with the invasion of Sargon the Great, the leader of the Semitic Akkadians.

Under him the indigenous Sumerian and the invading Akkadian cultures were merged. Sargon’s empire stretched from the Black Sea in the north to the Persian Gulf in the south. Consequently a considerable amount of information sharing subsequently took place. Thereafter followed Hittite, Asyrian and Persian invasion. All seemed to join the existing culture and the strong cultural unity remained. The use of the cuneiform script formed a strong bond between the peoples and today we have a huge collection of the baked clay tablets bearing a plethora of documents ranging from laws and school lessons to stories and personal letters.

Tablets dating from around 1700BC indicate a well-established number system. The Babylonians had taken the decimal (base10) system, used by many cultures around the world at that time, and integrated it into a sexagesimal (base 60) system. A measurement of sixty units can be divided into halves, thirds, quarters, fifths, sixths, tenths, twelfths and fifteenths more easily than a measurement of 100 units. So it appears that the sexagesimal system had been adopted to facilitate the subdividing that accompanies measuring.

The efficacy of the Babylonian sexagesimal system is apparent by its survival into the twentieth century, in the form of our current measurements of angles and time. The Babylonian numbers are shown in the attached appendix. To complement their numbers, the Babylonians developed a place value system, an extraordinary achievement, some 4000 years ago. They then saw that they could use there developed “digits” in columns representing 600, 601, 602, etc to represent any number.