Welshman invents the '='sign
Robert Recorde 1512-1558 – Britain’s leading medieval mathematician.
Robert Recorde was born in Tenby in Wales, then a busy fortified trading town where he could observe traders working out the cost of deals using beads on a grid (like an abacus). He also watched them using hand signals to represent numbers like traders used to do on the floor of the world’s stock exchanges. He went to Oxford and Cambridge for his education, qualifying as a medical doctor which became his main source of income, but he was also interested in theology, the sciences, astronomy and especially maths. He lived through a period of Catholic/Protestant division (Henry VIII, Edward VI, Mary) and had official positions in charge of one of the royal mints and of mining operations in Ireland (for lead and silver). He was a Protestant and always had to be mindful of offending those in power. Eventually his enemies had him arrested but, like many, he died of fever in prison (probably dysentery or typhus) before allegations of his mismanagement of mining in Ireland (and if cleared of that perhaps of heresy) could be looked into; at least he avoided being burnt at the stake!
I am going to focus on his mathematics. He wrote one of the first maths books published in English. It was titled The Grounde of Artes (1543) and explained the uses and methods of arithmetic. It sold very well and new editions were produced at least 36 times (6 of them in Recorde’s lifetime). At that time it was mainly scholars and clerics in the universities and monasteries that had access to, and could read, books in Latin and Greek. He wanted to spread knowledge about maths in his own language. Most people knew words for numbers and could add simple small numbers, but multiplication was known only to a few. Some would know the products up to about 5 x 5.
The book had over 300 pages and was laid out like a dialogue between Recorde as the master and a student. Often the student appears not to understand at first and the master explains things in different ways to help the student (and the reader). He regularly praises the student for getting the right answers, giving the reader, too, confidence that he is making progress (a good teaching technique).
His book explains a method for bigger products than 5 x 5, e.g. 8 X 7
1) Write down the two numbers in a column 2) Next to them, put the numbers that make the total in that row 10
3) Multiply the two small numbers in the right column and put the answer below.
4) Take the difference of either diagonal pair e.g. 7 – 2 = 5 or 8 – 3 = 5
and put this answer in the left column
5) The answer is 8 x 7 = 56
You can adapt the method for multiplication of bigger products. e.g. 87 x 96 e.g. 993 x 886
87 13 993 7 96 4 896 104 83 52 so 87 x 96 = 8 352 889 728 so 993 x 896 = 889 728
YOU: All years: Try the method with a few other examples e.g. 6 x 8, 93 x 94, 75 x 96, 992 x 974, noting where an extra 0 is sometimes needed or to allow for a carry. Yr 10-13: Explain using algebra why the method works.
Recorde covers, in the Grounde of Artes, written methods for addition, subtraction, multiplication and division and gives lots of examples of using arithmetic as motivation for learning the written methods.
e.g. working out bills for goods from a merchant, wages ( so many days/hours at so much per day) expenses ( rent + food + services)
His examples are not always easy. Here are two taken straight from the book. Problem ① If I sold you a horse, having four shoes, and in every shoe there were six nayles, with this condition, that you shall pay for the first nayle 1 ob, for the second nayle 2 ob, for the third 4 ob, and so forth doubling unto the end of all the nayles. Now I ask you, how much would the whole price of the horse come unto?
Money in 1543 came in more denominations of coins than we use: 1 pound = 20 shillings, I shilling = 12 pennies, 1 penny = 2 half-pennies = 4 farthings. The abbreviations in the book for these were pound = li, shilling = s, penny = d, half-penny = ob, farthing = ö. There were other coins too (sovereigns, royals, nobles, crowns, groats, kews and cees); another rich source of conversion sums.
YOU – Yr7-11 – try to answer question ① – messy but straightforward. Probably easiest to work in obs throughout and then convert the total at the end to a sum of money in li, s, d, ob. Yr 12-13 – use your knowledge of geometric series to get the answer more quickly.
The answer to problem ① is a very large sum of money and Recorde may have heard of similar problems from other cultures – perhaps from India where there is a 14th century Hindu legend that the disguised Krishna played chess with a wise ruler. Krishna won and as reward asked for one grain of rice on the first square of a chess board, two on the second square, four on the third and so on. The ruler thought he was getting off cheaply but his clerks worked out that the total was massive and he could never pay his debt. Krishna lets him off his debt as long as the ruler provides free rice for his poor people for the rest of his life Problem ② A lord delivered to a bricklayer a certain number of loads of bricks, whereof he willed him to make twelve walls of such sort that the first wall should receive two thirds of the whole number, and the second should receive two thirds of that, that was left, and so every other two thirds of that that remained, and so did the bricklayer. And when the twelve walls were made, there remained one load of bricks. Now I ask you, how many loads went to every wall, and how many loads was in the whole?
Problems ① and ② give lots of practice at multiplying and adding and using money. In problem ② he meant his readers to start with the 1 load left after the twelfth wall and so deduce that 2 loads were used for the twelfth wall. That means 3 loads were left after the 11th wall, so the 11th wall used 6 loads,……. Keep going to the first wall and add up all the loads for the 12 walls. YOU: yr 7/8 Use Recorde’s suggested method to solve problem ②. yr9-11 Use Recorde’s method and then start again and use fractions to work out what fraction the 1 load left at the end is of the whole and so work out the total number of bricks without working out the individual numbers of loads in the 12 walls. yr12-13 Use your knowledge of geometric series to write down an expression for a, the number of loads for the first wall, and hence work out the total number of loads and a formula for the number of loads in any of the walls.
In a second maths book, The Whetstone of Witte (1557), Recorde extended from his first book and explained about solving equations. He talks about surds (numbers, he says, with infinite decimal places) and invented and used for the first time the equals symbol “=”, now ubiquitous throughout maths. The symbol he invented for equality in equations he called a pair of parallels, or gemowe lines (gemellus = latin for twin lines), thus, , because “no two things can be more equal”.
Not only the equals sign, but all symbols were elongated compared to modern practice. There was no way at that time of writing powers of x so Recorde used different symbols for each of the powers of x, such as x², x³,…...
Converting this into modern notation he gives the following example of an equation (which is no 6 in the cover photo):
YOU Yr 10+: Solve the equation Recorde quoted. It has two rational roots.
Recorde also published a book on geometry in English and another on astronomy. His book on astronomy was written under the current religious belief that the earth was the centre of the universe. His book was ready for publication when he heard in a letter about the ideas of Nicolaus Copernicus. He still published his book, for there was much in it still of practical use to navigators, but Recorde had the grace to acknowledge in his book the great new ideas that Copernicus had just announced that put the sun at the centre of the universe, when so many others just ignored the new theory. He probably realised that Copernicus was right, as careful observations by Tycho Brahe, and calculations by Johannes Kepler demonstrated 50 years later.
Main Sources: Robert Recorde by Gordon Roberts (UWP 2016) The History of Mathematics by Carl Boyer (Wiley 1968)
Answers to the problems will be in my next post. GDI April 2020