According to the Oxford English Dictionary, mathematics has two main aspects: pure; and applied. The pure form is an 'abstract science of number, quantity and space studied in its own right'. The applied form is 'as applied to other disciplines such as physics, engineering etc'. The dictionary also gives a definition as 'the use of mathematics in calculations ...' The span is enormous, without limit, much as is music, and literature with their 'pure' forms needing no justification in terms of usefulness in the real world; each is pursued for sheer delight. The applied form is a necessity if physics, engineering and all other practical sciences are to progress; these disciplines rely on the acquisition of data from experiments and the data are used in calculations relating to formulae, which are symbolic representations of certain features of what is being studied. For instance, if the study in question is the lengths of the sides of a triangle, one case is summarised famously in a theorem (statement) attributed to Pythagoras: in a triangle with one angle (the right angle) of 90 degrees, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares on the other two sides. While this is a precise expression it is rather cumbersome for the purpose of calculation. A neater form is an algebraic (symbolised) one, namely a² = b² + c² , where a² is the length of the hypotenuse and with the index 2 meaning ’multiplied by itself’. The smallest digits that confirm the theorem are 3, 4, 5. This is an obviously use of mathematics given the profusion of right-angled triangles in our man-made environment, but what if none of the angles is 90 degrees? Is there still a formula for the sides? This is where study 'in its own right' begins to appear. And, in the wake of knowledge of square, what about the opposite process, namely, square root. It is easy, by mental arithmetic, to know that 5 is the square root of 25, but what is the square root of 26? Is there such a number with the same preciseness as 5?

A famous name in geometry (the study of points, lines, surfaces and solids) is that of Euclid. One of his famous statements is often represented as 'the shortest distance between two points is a straight line'. It is often unaccompanied by 'on a plane (flat) surface'. Do the same rules/outcomes apply if the surface is spherical? This question is crucial for navigation.

These instances serve to illustrate some of the simpler variations in the panoply that is represented by the single word 'mathematics'. A little digression: in today's educational turmoil is it sensible to use this simple word to represent what is a compulsory study for all pupils up to 16 years of age? When industrialists and business representatives complain (as they have done for 50 years to this author's knowledge) about standards of mathematics of pupils leaving school at 16+, are they really complaining about the vast range of the discipline. Perhaps the term for general education should be 'transactional mathematics'.

The following examples illustrate the contributions of some mathematicians of Wales.

His place of birth is known to be Tenby but its date is uncertain. His mother was a native of Machynlleth. He went to Oxford University in 1525 and became a Fellow of All Souls' College in 1531. From Oxford he went to Cambridge where he qualified in medicine in 1545. In due time he became physician to the royal family and moved in distinguished, privileged society circles. However, he got embroiled in and lost a libel case brought against him by the Earl of Pembroke. The outcome was that he was sent to Southwark prison where he died in 1558.

He wrote several significant books on arithmetic and geometry. In these he promoted the use of mathematics symbols, to a greater degree than previous authors. He is best known today for his introduction of the equals sign =. 'I will sette as I doe often in woorke use, a pair of paralleles ... bicause noe 2 thynges can be moore equalle'. Outside mathematics he shared his interests in astronomy, being among the first to promote the views of Copernicus. Based on his training and practice in medicine he summarised his experience in another book, The Urinal of Physick (1548).

William Jones (1675–1749)

William Jones the mathematician, by William Hogarth, 1740

He was born in a small village in Anglesey and attended Llanfechell Primary School. Here his prowess in mathematics was recognised by a local landowner, Lord Bulkeley who arranged for him to go to London for training and employment.

His employer in London, a merchant, arranged for William to go to the West Indies, an experience which matured an interest in the mathematics of navigation. On his return to London he married 'well' twice and moved easily in the upper echelons of London society. He wrote several influential books on mathematics. Such was his distinction that he became Vice-President of the Royal Society and was appointed to the committee that was established to determine whether Newton or Leibnitz should be recognised as the author/originator of the differential calculus, a crucial aspect of mathematics to do with rates of change.

Perhaps his most significant contribution to mathematics was his use, before anybody else, of the Greek symbol (pi) for the ratio perimeter/diameter of a circle. Very many people will recall from their schooldays that ∏ can be represented as 22/7 or 3.142... This is a very interesting number in pure mathematics, an irrational number, one that cannot be represented precisely as the ratio of two integers.

He was born in Staylittle, Powys (a few miles from Llanidloes) and received his early primary education in the village school, almost exclusively in Welsh. In 1896 the family moved to Llanhilleth in Monmouthshire, where he continued his primary education before moving to the Abertillery Intermediate School in 1899. His mathematical abilities were very distinguished and enabled him to win a scholarship to the university in Aberystwyth. His career was mostly in meteorological research, initially with an examination of periodicities in European weather, is a shining example of applied mathematics. An early example on the dynamics of cyclones and anticyclones is typical of his gift for reducing a problem to its essentials and extracting useful conclusions with the minimum of calculation.

Gwilyn Meirion Jenkins (1932-1982)

He was educated at Pontlliw Primary School and Gowerton Boys Grammar School, both near Swansea. His ability in mathematics was soon evident. He specialised in the subject in London University. His particular interest lay in the application of mathematical principles to the study of the behaviour of 'systems', mostly in an industrial, engineering context. He became, at the age of 35, the first Professor in the first Department of Systems Engineering to be established in a British University, at Lancaster.

His emphasis, throughout his all-too-short career, was on systems engineering as the study of complex systems in their totality, so as to ensure that they achieve their overall objectives as efficiently as possible. By today, these studies are widespread through industry and society as a whole, the best illustrating the richness of Jenkins' foresight.

Professor Kenneth Walters, MSc PhD DSc (Wales) FRS (born 1934)

A native of Swansea he was a pupil at Terrace Road Primary School and Dynevor Grammar School. He continued his studies at the university in Swansea, specialising in Applied Mathematics. After spending a year (1959-60) in the USA, Ken took up a position on the staff of the mathematics department at Aberystwyth where he remained for the rest of his career, retiring as professor. His distinction led to his being elected a Fellow of the Royal Society in 1991.

His specialist field is rheology, which is the study of the mathematical basis for the technology used to measure the flow properties of fluids. Over many years of distinguished leadership he established new procedures for studies that proved extremely challenging for conventional mathematics. To this end he pioneered procedures using computers and lasers.

David Rees (1918-2013)

He was born and brought up in Abergavenny and attended the local secondary school, King Henry VIII Grammar. His mathematical talent was evident from an early age and culminated in a first-class degree from Sidney Sussex College at Cambridge University. As with many other distinguished scientists of all disciplines, David's career was interrupted by the Second World War. He was seconded to Bletchley park, the code-breaking centre in Buckinghamshire. Here he worked here for the duration of the war, after which he took up an assistant lectureship at Manchester University and later to a fellowship at Downing college at Cambridge University. David left Cambridge in 1958 to become professor at Exeter University where he remained until his retirement in 1983.

His specialist field was commutative algebra which can be loosely described as the algebra underlying the equations used to describe curves, surfaces and higher-dimensional geometric objects. This can be safely described as mathematics of the 'pure' kind to which Rees made a very distinguished contribution that still informs current studies.