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Science and technology: Sample feature essay
The Origins of Mathematics
by Peter Higgins
First mathematics
It is often said that people first encountered mathematics when they began to count their livestock or tried to measure the size of a field. The two sides of the mathematical coin, discrete mathematics based on counting, and continuous mathematics that arises
through measurement, still form the basis of modern mathematics.
Inventing names for numbers
Counting and arithmetic did not come about easily for a variety of reasons. To count we just need tally marks; but in order to talk about counting, we need a name for each number we use. Every language combined names of small numbers as a way of expressing larger ones, as in the French word for 80, quatre-vingt (four twenties). The ancient Greeks used letters to represent numbers so that  was 1 while  stood for 20 and in that way would write for 21. They could equally have written to convey the same meaning, one-and-twenty.
Roman numerals were based on ten with the basic symbols being I, X, C, and M for 1, 10, 100, and 1,000 respectively, although they also introduced V to stand for 5, L for 50, and D for 500. The symbols were generally written in descending order, so that
1,944 = MDCCCCXXXXIIII. Sometimes they made use of position: a smaller unit placed before a larger one indicated subtraction of the smaller from the larger – for instance 9 was written as IX instead of VIIII. So 1,944 = MCMXLIV. But this representation is not always as easy to understand or employ in arithmetic, which may be why the Romans did not always make use of it.
Positional systems and '0'
In our number system, unlike the Greek system, order matters. Take 21. Swapping the places of the numerals 2 and 1 gives 12, a different number, for the 1 now represents 1 ten, while the 2 means two units.
No ancient European society devised a complete positional numbering system in which the meaning of a numeral depends on its position within the number and full use is made of a zero symbol. The idea of a zero symbol was used by the Mesopotamians and Babylonians, and was employed in the way that we do to distinguish between 74 and 704. The full potential of the system was not embraced, however, as the 0 was seldom used in the final place, the way we show the difference between 74 and 740.
There were nonetheless many practical and sophisticated counting systems in the ancient world. Commercial and trading societies often constructed good systems of arithmetic and the peoples of ancient Mesopotamia did have a sexagesimal positional system, one based on 60, over four thousand years ago. Numbers exceeding 60 were written according to the positional principle, while combinations of the symbols from one to ten were used to make the basic numbers of their system which ran from 1 to 59. For instance, the ancient clay tablets reveal examples like:
524,551 = 2 ×603 + 25 × 602 + 42 × 60 + 31.
The first complete positional system came into use in India around the 1st century AD. The symbol for 0 was called sunya, the Hindu word for 'empty'. It is the basis of our number system, called Hindu-Arabic, as it passed to Europe via medieval Arabic scholars. A positional system, complete with zero, was also invented by the Mayan civilization of Central America by the 6th century AD, based on multiples of powers of 20 instead of 10.
Early computing
Throughout Asia and Europe, arithmetic was carried out on the calculator of the ancient world, the abacus (Greek 'sand tray'). The main obstacle to written arithmetic was lack of cheap writing materials. The first example of long division is by Calandri in 1491. The decimal system of fractions did not firmly take root until planted by force after the French Revolution of 1789.
A typical abacus consisted of a wooden rectangular frame in which a series of parallel rods were housed. Along each were a number of identical beads. Cutting across the rods was a counting bar. One rod represented the units column and the beads on rods to its left each represented multiples of 10, 100, 1,000, and so forth. Beads above the counting bar counted for five while those below represented one. Addition using an abacus is easy, as we need only count the number of each bead type and carry over to higher units as the need arises, although to carry out subtractions may require borrowing from the next highest rod.
A great advantage of the pen and paper methods that emerged in the Renaissance is one of communication, for they allow working to be shown and checked. The scribes of the ancient Babylonian tablets left descriptions of numerous problems and their answers, but we would need to see the clerks of the ancient world in action on their counting frames to appreciate exactly how they did their sums.
Geometry and paradox
An early use of geometry and measurement arose in Egypt where the ancient Greek historian Herodotus tells us that the Nile's annual flood regularly washed away boundaries and landmarks so that a system of accurate surveying was needed in order to reaffirm who owned what: indeed the Greek word 'geometry' means 'Earth measure'. The founder of geometry was the Greek philosopher and scientist Thales of Miletus, who is said to have impressed the Egyptians by measuring the height of the Great Pyramid of Cheops
through use of shadows.
His successor was the Greek mathematician and philosopher Pythagoras of Samos, best known for his theorem that says that the square on the longest side (hypotenuse) of a right-angled triangle has an area equal to the sum of the area of the squares of the other two sides. Pythagoras is also said to have discovered irrational numbers. If a right-angled triangle has two sides which are both 1 in length, the hypotenuse will be the square root of 2. Pythagoras proved this number was irrational, that is to say that it cannot be represented by any ordinary fraction a/b. Up to this point, it was taken as self-evident that, in principle, any constructed line could be measured exactly using a standard ruler, provided that we marked the ruler with a sufficiently fine scale. Pythagoras had proved this to be false.
This and some other paradoxes in classical mathematics were eventually resolved in the 4th century BC by the Greek mathematician and astronomer Eudoxus with his Theory of Proportions, an account of which is to be found in The Elements, the classical texts written by the Greek mathematician Euclid of Alexandria. Eudoxus introduced a theory that applied equally well to all lengths by making subtle use of inequalities to deal with equalities.
Later achievements
Euclid's Alexandrian School remained the leading centre of thought during the later classical period. Its greatest genius was the Greek mathematician Archimedes, who took both geometry and mechanics to new heights. He died in 212 BC, probably killed by the Roman invaders of his home city of Syracuse, which he brilliantly defended through the use of devastating war machines that capsized the vessels of the invaders. In his tract, The Method, Archimedes allied mathematics and physics by insisting on rigorous standards of proof while emphasizing the importance of physical intuition as a guide to the truth.
The Greek mathematician Apollonius of Perga, a younger contemporary of Archimedes, gave the definitive description of curves
arising from cones that proved to be a major ingredient of the theory devised by English physicist and mathematician Isaac Newton nearly two thousand years later to explain planetary orbits. The Greek mathematician and engineer Hero of Alexandria (lived AD 62) invented the first working steam engine, made a primitive thermometer, and proved the formula for the area of a triangle in terms of its three sides. Other outstanding figures were Greek geographer and mathematician Eratosthenes, who calculated the diameter of the earth in 230 BC through the difference in the sun's elevation at Syrene and Alexandria at the summer solstice, and Greek mathematician Diophantus, whose treatise on number theory inspired French mathematician Fermat's last theorem, while The Collection of Greek mathematician, astronomer, and geographer Pappus was the final great intellectual work of classical times.
Decline and fall
Despite its success and staggering sophistication, Greek mathematics remained wedded to the geometric style of Euclid in which even common algebraic facts about numbers were demonstrated in what appears to us a strange and unnatural fashion through areas of
geometrical figures. Even the Ancient Babylonians, one thousand years before the birth of Pythagoras, seemed more at home with algebra. Although they did not use symbols to stand for numbers as we would, they demonstrated how to solve quadratic equations (in which the unknown quantity x appears through its square, x2) and compiled astonishingly extensive tables of number triples such as (3, 4, 5) and (4961, 6480, 8161) where the sum of the squares of the two smaller numbers equals the square of the larger. The precursors of modern algebraic methods are to be found in societies such as Mesopotamia, India, and China.
After the 4th century AD classical Greek mathematics entered terminal decline. Mathematics was not revived until the dawn of the modern age when the need to solve problems in navigation and the physical world ushered in a fresh and progressive epoch.
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