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      Ishango bone

      The earliest mathematical thought was also concerned with magnitude — the size of an object compared to other objects of the same kind. To Palaeolithic hunter-gatherers, how many objects there were and how big they were was an important piece of survival data, as was form. This refers to the configuration of an object, its visual appearance, and, basically, its shape.

      Recent studies of animal intelligence have reached very interesting conclusions about the basic levels of mathematical ability of this elementary type that some animal species seem to share with human beings. Numerous “counting” dogs and horses have featured as circus and vaudeville acts, and they certainly seem to show some basic number skills.

      An impressive university study on animal mathematical ability was conducted by Dr. Naoko Irie in Tokyo. Elephants from the Ueno Zoo watched as apples were dropped into buckets, and the elephants were then offered their choice of the buckets. Human subjects were also involved in the experiment to compare their results with those of the elephants. The elephants scored 74 percent while the human beings scored only 67 percent. The experiment suggested that when more than a single apple was dropped, the elephants had to carry out the equivalent of running totals in their heads.

      The history of mathematics indicates that as civilizations developed, the demand for mathematics increased. The old commercial civilizations, such as Sumer in the region of the Tigris and Euphrates, needed to make careful records of commercial transactions: jars of oil, measures of corn, units of cloth, slaves and animals bought and sold. The Sumerians developed writing, irrigation, agriculture, the wheel, the plough, and many other things. Their writing system, known as cuneiform, used wedge-shaped characters cut into clay tablets that were then baked. As a consequence, they have lasted thousands of years and archaeologists have studied them closely for centuries. In the Sumerian civilization there was the need to measure areas of land and to calculate taxes. Sumerians developed calendars and were keenly interested in observing and recording the stars and planets in their courses. They developed the use of symbols to represent quantities. A large cone stood for “60.” A clay sphere stood for “10,” and a small cone was a single unit. In addition to these developments, they used a simple abacus.

      Just as the popular base-10 decimal system of numbering is almost certainly based on the fact that we have 10 fingers, so it is suggested that the Sumerian and later Babylonian sexegesimal system (base-60) is based on the 12 knuckles of 1 hand and the 5 fingers of the other, which create 60 when multiplied together. Five hands would be thought of as containing 60 knuckles.

      This base-60 system had many advantages. For example, “60” is the smallest number into which all numbers from 1–6 will divide exactly. The number “60” is also divisible by 10, 12, 15, 20, and 30. The convenience of “60” can still be seen in the concept of having 60 seconds in a minute, and 60 minutes in an hour. The 360 degrees of a circle is based on 60 multiplied by 6.

      The Babylonians also used an early version of the “0,” although they seem to have employed it more as a place marker than as a symbol representing nothing. Five thousand years ago the Sumerians and Babylonians were making complicated tables filled with square roots, squares, and cubes. They could deal with fractions, equations, and even algebra. They got as close to π as regarding it as 3 1/8, or 3.125, which isn’t far from our contemporary 3.14159….

      They also had the square root of 2 (1.41421) correct to all 5 decimal places. The square root of 2 is very useful for calculating the diagonal of a square. The formula is:

      side of square×√2=the diagonal of that square

      As a maths tutor, co-author Lionel passes that useful shortcut to his students along with the square root of 3 multiplied by the side of a cube to calculate the diagonal of a cube. The formula is:

      √3×side of cube=diagonal of cube

      Other Babylonian tablets provide the squares of numbers up to 59 (59×59=3481): a major achievement for mathematicians without calculators or computers!

      The rich leisure culture of Babylon had numerous games of chance, and the dice they designed for these provided further archaeological evidence of their mathematical knowledge. This would seem to suggest an area of early thought where mathematics and numerology share the territory. Gamblers enjoy using systems of “lucky” numbers to try to beat the odds. In the old Babylonian games of chance, players may well have played their luck with numbers that they hoped would prove to be influential in moving the odds in their favour. Outstanding mathematicians like Marcus du Sautoy have examined these theories and suggested among other things that picking consecutive numbers can increase a gambler’s chances of winning a lottery.

      Their buildings were also geometrically interesting, and the Sumerians and Babylonians had no problems calculating the areas of rectangles, trapezoids, and triangles. Volumes of cuboids and cylinders were also well within their mathematical capabilities.

      One of many interesting problems in the history of mathematics and numerology is the famous Plimpton 322 tablet. It came from Senkereh in southern Iraq, and Senkereh was originally the ancient city of Larsa. The tablet measures 5 inches by 3.5 inches, and was purchased from Edgar J. Banks, an archaeological dealer. In 1922, he sold the mysterious tablet to George Plimpton, a publisher, after whom it was named. Plimpton placed it in his collection of archaeological treasures and finally bequeathed them all to Columbia University.

      Written some 4,000 years ago, the tablet contains what seem to be Pythagorean triangle measurements — written centuries before Pythagoras lived! The classical “3, 4, 5” Pythagorean triangle is right-

       angled because 32+42=52, also expressed as 9+16=25. Any triangle

       with those ratios will also be right-angled, for example “6, 8, 10” produces 36+64=100. Whoever carved the Plimpton 322 tablet millennia ago seems to have been well aware of that.

      Evidence for the development of mathematics in ancient Egypt was found in a tomb at Abydos, where ivory labels with numbers on them had been attached to grave goods. The famous Narmer Palette was discovered in 1897 by J.E. Quibell at Hierakonpolis, the capital of Predynastic Egypt.

      The palette reveals the use of a 10-base number system and accounts for thousands of goats, oxen, and human prisoners. Inscriptions on a wall in Meidum, near one of the mastaba (a low-lying, flat-bench-shaped tomb), carry mathematical instructions for the angles of the walls of the mastaba. These inscriptions involve the cubit as the unit of measurement. This was a unit based on the size of parts of the body, from the elbow to the fingertips, approximately 18 inches.

      The Rhind Mathematical Papyrus, which is well over 3,500 years old, was acquired by Alexander Henry Rhind in Luxor in 1858 and is now in the British Museum. It was copied from a much older papyrus by a scribe named Ahmes, who described it as being used for “Enquiring into things … the knowledge of all things … mysteries and secrets.” It would seem from this that Ahmes had a numerological view of the power of numbers. They were not merely of use to solve mathematical problems: they had magical powers as well.

      The Rhind Papyrus contains a number of problems in both arithmetic and algebra, which has led some antiquarians to suggest that it was perhaps intended as a teaching document. One interesting example shows how the Egyptian mathematicians of that period took the numbers from 1–9 and divided them by 10. They worked out that 7÷10 could be expressed as 2÷3+1÷30. The papyrus continues with interesting practical problems such as dividing loaves of bread among 10 men. There are then algebraic examples of linear equations such as x+1÷3x+1÷4x=2 in modern notation.

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