Welcome to issue 12 of the NCETM Secondary Magazine. Read on to discover our regular fortnightly features. Why not let us know what you want to see in forthcoming issues. If you have thoughts to share you can also add your comments on the portal.

Pascal – What’s in a Name?

The 19 June will be the 385th anniversary of the birth of Blaise Pascal. Students have usually heard the name Pascal either because of ‘Pascal’s triangle’ or because he gave his name to the programming language PASCAL. So, let’s get a few more details about Pascal and find out why he was so important.

Pascal was born in Clermont-Ferrand, France, in 1623. He was a child genius and educated at home by his father, who was also a gifted mathematician. The family moved to Paris, which provided the intellectual stimulus of the French Academy and allowed Pascal to attend meetings with such important mathematicians as Roberval, Desargues, Mydorge, Gassendi and Descartes.

Pascal’s early interest was in the field of geometry – he once said: “What goes beyond geometry goes beyond man.” He had studied Euclid and it is said that he drew on the floor with charcoal to rediscover some of Euclid’s theorems. Pascal, aged 16, wrote a treatise about the ‘Mystic Hexagram’ which became known as ‘Pascal’s theorem’.

Pascal succeeded in building an ‘arithmetic machine’ which used gears to add and subtract – this helped his father, who needed to balance his accounts every evening. It was called the Pascaline; only a small number (about 50) were ever made because it was so expensive. It is in recognition of this achievement that Niklaus Wirth named his high-level computer language ‘Pascal’ in the 1970s.

In 1653, Pascal wrote his 'Traité du triangle arithmétique' ('Treatise on the Arithmetical Triangle') in which he described the work he had done with a triangular array of numbers which had been used by Chinese mathematicians. We recognise the rows of Pascal’s Triangle as the binomial co-efficients.

Here is a linked problem – looking at the arrangement of letters below, how many downward paths can you take to spell ‘mathematics’?

It might be easier to break this problem down and do an easier one first – let’s do the same with ‘MATHS’.

You can see the six paths here – Pascal’s triangle could have helped with this problem. If you place the arrangement of letters over the triangle with the ‘M’ on the top 1, the ‘S’ will lie over the 6. When you have convinced yourself that this is true, you can try for the word ‘mathematics’.

Through a friend, Pascal became interested in the probability associated with gambling. His correspondence with Fermat on the topic led to the development of probability theory. Pascal recorded his idea, which became known as ‘Pascal’s wager’, in the book of thoughts or ‘Pensees’ that were published after his death. Pascal said that it was a safer bet to believe in God than not to believe:
- if God does not exist then there is nothing lost by believing in him
- if God does exist then there is everything to gain by believing in him.

Pascal was a religious man who, after a potentially fatal accident, devoted himself to theology and was influenced by Jansenism, a splinter group of the Catholic church. His sister became a nun in a Jansenist convent. Pascal’s interests were not confined to mathematics: he designed a public transport system for Paris, which was used in 1652, and his work with fluid systems led to the development of hydraulic systems such as the syringe.

In his 39 years, Pascal did an incredible amount which has had an influence on the mathematics we use today. In the new Key stage 3 Programme of study it says:

Mathematics has a rich and fascinating history and has been developed across the world to solve problems and for its own sake. Pupils should learn about problems from the past that led to the development of particular areas of mathematics, appreciate that pure mathematical findings sometimes precede practical applications, and understand that mathematics continues to develop and evolve.

So how will you celebrate Pascal’s birthday with your students?