For the Recorde: A History of Welsh Mathematical Greats

By Gareth Ffowc Roberts

This book celebrates the life and work of twelve mathematicians who were either born in Wales or who worked in Wales. When the Welsh national anthem was composed in 1856, Wales was at the centre of the industrial revolution, the country was transformed by engineering and technology, and scientific societies flourished across the length and breadth of the land. By 1859, Charles Darwin had published his On the Origin of Species, and one of its outcomes in Wales was a growing tension between religion and science, which influenced peoples’ perceptions of their Welshness. By the end of the nineteenth century, that perception had narrowed to include its poetry, music, religion and little else. Following the popularity of his book Count Us In, the author adopts a similar style inviting us to take pride in our mathematicians and demonstrating how the tide has turned.

• Language: English
• Publisher: University of Wales Press
• Release date: Sep 15, 2022
• ISBN: 9781786839183

Book preview

1 THINK OF A NUMBER

Tenby

Robert Recorde, 1512?–58

This bust, dating from 1910, is based on a painting of Recorde now housed in the University of Cambridge. It has been established that the painting is by a seventeenth-century Dutch artist. This is certainly not Recorde, but the image continues to be used to represent him in the absence of anything more authentic.

The bust of Robert Recorde in St. Mary’s Church, Tenby

The first equation ever to appear in print in which the equals sign is used (The Whetstone of Witte (1557))

Have a go at these three puzzles:

I’m thinking of a number. Double the number is 8. What’s my number?

I’m thinking of another number. Adding 5 to it, I get 12. What’s my number?

I’m thinking of a third number. Adding 5 to double the number, I get 27. What’s my number this time?

How did you get on? How did you get your answers? Some find it easier to picture the missing number in a box or hiding behind a cloud and ‘see’ the first puzzle ‘in their heads’ something like this:

double is 8. What number is in the box?

or

double is 8.

What number is hiding behind the cloud?

Does that help?

The third puzzle could look something like this:

double add 5 is 27.

What number is hiding behind the cloud this time?

A further step, which is quite a leap, is to use a symbol to represent the missing number. Traditionally, the symbol used most often is the letter x and the puzzles could then look like this:

I’m thinking of a number, x. Double x is 8. What’s x?

I’m thinking of another number, x. Adding 5 to x, I get 12. What’s x?

I’m thinking of a third number, x. Adding 5 to double x, I get 27. What’s x this time?

Notice that I’ve used x in each puzzle. The choice of symbol is of no consequence: it could be a box or a cloud or x or y or an image of a cat or an elephant or anything else that springs to mind. Letters of the alphabet are commonly used and x and y are popular choices, although there is nothing special about either of them. Casting your mind back to your school days, and to algebra lessons in particular, it’s possible that the symbols x and y stand out. It’s also possible that you can recall the feeling quite suddenly of being lost as you were faced with things like this:

2x = 8. What’s x?

x + 5 = 12. What’s x?

2x + 5 = 27. What’s x?

The change from dealing with numbers to dealing with symbols can be an overwhelming conceptual leap unless school children (and adults) are introduced to the use of symbols with great care, step by step.

Can you sympathise with the feeling expressed in this message?

Dear Algebra

Please stop asking us to find your x.

She’s never coming back, and don’t ask y.

But there’s nothing to fear. Think of x as the number in the box or the number hiding behind the cloud and the challenge is to find out what number it is. That, in a nutshell, is the essence of any and every equation.

Algebra isn’t a recent invention. Some of its earliest ideas were developed more than 3,000 years ago by the Babylonians and later by the Greeks. The modern word ‘algebra’ is derived from the Arabic al-jabr, one of the techniques used by the Arabic mathematician al-Khwa¯rizmı¯ in about AD 800. Born in the Middle East in what is now Uzbekistan, al-Khwa¯rizmı¯ was an astronomer and a geographer as well as a mathematician. He worked and studied in an important and influential centre of learning in Baghdad. Much has changed since then.

Robert Recorde was caught up in the whirl of the Renaissance movement in Europe in the sixteenth century, rediscovering the riches of the classical age and the developments in the Middle East, including the work of al-Khwa¯rizmı¯. One of Recorde’s feats was his contribution to the development of algebra in a very special way.

Robert Recorde was born in Tenby, Pembrokeshire, probably in 1512. It was there, as a small boy, that he first became enraptured by numbers and mathematics generally. After graduating in mathematics at Oxford and in medicine at Cambridge, Recorde was employed by the Crown to oversee the work of the royal mints in Bristol, Dublin and London, as well as the silver mines in Ireland. His immediate ‘line manager’ was the Earl of Pembroke, a member of the Privy Council. Recorde was an honest and responsible person, but he lacked the political guile needed in a period of deceit and intrigue. He accused his master, Pembroke, of malfeasance by diverting part of the profits made by the mints into his own pocket. Recorde was accused of libel by Pembroke and was duly hauled before the courts to answer his accuser. Pembroke’s political might was too great: Recorde lost the case and was fined a thousand pounds. He lacked the means to pay such a vast sum and was incarcerated in a debtors’ prison in London. Within a few months he had contracted a disease at the prison, where he died in 1558. Some years later, during the reign of Queen Elizabeth, the case was reopened and Recorde was exonerated. It was too late by then, of course, to compensate Recorde, but his family was given land in Tenby to secure their well-being.

Recorde’s most significant legacy was in the richness of the books on mathematics that he published. He had thought long and hard about how best to present mathematical ideas to both children and adults, particularly to those who had not benefited from a classical education in Latin and Greek. He was the first to write books on mathematics in English, to ensure that they could be understood by the general population of Britain, those referred to by Recorde as ‘the vnlearned sorte’. He set a standard that was not bettered for more than 300 years. In effect, Recorde was the first teacher of mathematics in these islands.

Recorde is best known for his invention of the familiar equals sign, ‘=’, a symbol that appears in all the equations in this book. The symbol is now so commonplace that it is taken for granted, but its introduction was a major step in the development of mathematics.

The first equation written by Recorde in his book on algebra, The Whetstone of Witte, looks distinctly odd to our modern eyes, more than 450 years later. Don’t be put off by it:

The first equation, from Robert Recorde’s book,

The Whetstone of Witte (1557)

It is instructive to linger over this equation as it represents the efforts of Renaissance mathematicians to combine various symbols in order to simplify their work. The equation expresses something that is very familiar to school children today but it is presented very differently. This is how it would be written now:

14x + 15 = 71

The challenge is to work out the value of the symbol ‘x’. Treat the equation as a puzzle: I think of a number, multiply that number by 14, then add 15 to get the answer 71. What was my original number? You will quickly see that the number 4 fits the bill. Using modern jargon, the value of x that satisfies the equation 14x + 15 = 71 is 4.

But what are the other symbols that are in Recorde’s original equation? Some are immediately recognisable, while others are distinctly odd. The oldest symbols in the equation are very familiar to us and represent numbers 14, 15 and 71. These are Hindu-Arabic numbers that date back to about AD 600. These numbers travelled from India through the countries of the Middle East, across north Africa, over to Spain and on to the rest of Europe, reaching Britain towards the end of the fifteenth century – a long and slow journey. Before then, Roman numbers, such as clviii (for 158) and mmdcxxix (for 2,629), were used. In his book on arithmetic, The Ground of Artes, published in 1543, Recorde begins by introducing the ‘new’ numbers to his audience as follows:

And fyrste marke that there are but .x. figures, that are vsed in arithmetike, and of those .x. one doth sygnifie nothing, which is made lyke an o … The other .ix. are called sygnifienge figures, & be thus figured:

1 2 3 4 5 6 7 8 9

And this is theyr valewe.

  ii.   iii.   iiii.   v.   vi.   vii.   viii.   ix.

In effect, Recorde had to introduce the ‘new’ numbers to his readers by explaining them using the ‘familiar’ Roman numbers.

Another familiar symbol in the equation is the ‘+’ sign, representing addition. The shape of the sign in the equation differs slightly from the modern ‘+’ but it is easily recognisable nevertheless. The symbol was first used in Germany and Italy towards the end of the fifteenth century. Before then, words were used to convey the idea of addition, particularly the Latin et (for ‘and’), and it’s possible that ‘+’ developed as an abbreviation of et by concentrating on the crossed ‘t’. This symbol together with ‘—’ for subtraction were introduced to Britain by Recorde in 1557 in his book on algebra: ‘There be other. 2. signes in often vse, of whiche the firste is made thus + and betokeneth more: the other is thus made — and betokeneth lesse.’

But what can we make of the symbols and that are very strange to our modern eyes? Recorde was familiar with the work of European mathematicians, particularly those who wrote about their work in Latin, a language that acted as a lingua franca across Europe. He had particularly enjoyed books written by the German mathematician Johannes Scheubel (1494–1570) and used some of Scheubel’s ideas in his own book on algebra. One of Scheubel’s books is among the treasures in the National Library of Wales in Aberystwyth, the original owner of the book having written notes in Latin in the margin while browsing through the book. Ulrich Reich, a German mathematician who has studied Scheubel’s influence on Recorde, has suggested that Recorde himself may have been the book’s owner and that these words are a rare example of his handwriting:

A note in the margin of Scheubel’s book.

By permission of The National Library of Wales (b51 P(3)7)

In this note, the reader is thinking aloud while grappling with the significance of the text, the first word, Semper (‘always’), showing the mathematical mind at work trying to get to the nub of the argument.

What do we know of the symbols themselves? The symbol signifies that the preceding number stands alone, while the symbol represents an unknown quantity, the ‘x’. They both form part of a collection of symbols invented by German mathematicians as they tried to express algebraic ideas. They are referred to collectively as cossike numbers, a term derived from the Latin coss, meaning ‘thing or something’. The process of simplifying the whole system to the one with which we are familiar today was very slow.

The symbol that completes the equation is Recorde’s sign of equality. This is how Recorde explains what motivated him and how he set about