The evening before last was not a particularly exciting one on the TV so I tuned into a series on Channel 4 of which my son had spoken very approvingly. This was a four-part series of on the unfolding of the Suez crisis in 1956 in which Britain and France colluded with Israel in an abortive attempt to seize back control of the Suez canal, which had recently come under the control of Nasser the Egyptian leader. The programme was one of a four-part series of which I had already seen the first some time ago and now I was viewing the second in the series. The programme portrayed the way in which the military operation evolved almost hour by hour and was absolutely gripping so now I well understand why my son had been so enthusiastic about it on a first viewing. The Suez crisis put a an absolute end to whatever ambitions Britain had as a colonial power and, abandoned by the Americans, marks one of the worst mistakes in British foreign policy and Anthony Eden was forced to resign to give way to Macmillan shortly after its conclusion. I have another two episodes of this series to which look forward so they can wait until there is another poor night in the TV transmissions. After the watching of this documentary and the successful completion of a ‘fiendishly difficult’ Sudoku, a strange thought came into my head which was the proof of Pythagoras theorem (c²=a²+b²) from which derives the square root of ‘c’ as the length of the hypotenuse. Now I had learnt the mathematical proof of this when I was about 14 ad there was a reason why one had to be at this age before one was taught proof of the theorem. This is because one has to already have under one’s belt, as it were, certain elements of both geometry (for example the properties of similar and congruent triangles) as well as some algebra and then the two had to be brought together in one of the classic proofs of the Pythagoras theorem. I used to include an outline of this in one my lecture courses which was called ‘Thinking about Management’ and where I made the first part of a course ‘Thinking about Thinking’ and the proof of Pythagoras was used an illustration of both linear and logical thinking. Returning to the present day, I could not remember the proof of the theorem I had been taught when I was 14 years old but I know that it involved the dropping of a perpendicular and then using the ratio of the sides enclosing both the sine and the cosine angle before some additional algebraic manipulation. So I typed into my search engine the simplest proof of the Pythagoras theorem and soon accessed a simple illustrated visual which deployed the use of an off-set square within a square and the subtraction of the triangles to find the proof of the theorem. Now all of this sounds very complicated but was actually incredibly easier than the proof to which I had been subjected at the age of 14 in about 1959. The actual algebraic manipulation was minimal and the whole proof could be effected in about 2-3 lines of equations and I marvelled at the simplicity of it. I had imagined there was only one proof of the Pythagoras theorem but there are hundreds of ways of proving it and the actual method that was taught was not the proof that Euclid himself deployed but was formulated by the 12th century Indian mathematician Bhaskara who developed an elegant visual proof of the Pythagorean Theorem. I believe, but am not sure, that this very simplest proof is the one that is used in modern day GCSE mathematics calculations, the more classical proofs proving to be a too complicated for modern generations of school children. Not one of the students that I taught could prove Pythagoras’ theorem by the method by which I had been taught. The history of the theorem is fascinating, though. The earliest recorded knowledge of the Pythagorean theorem dates back to ancient Babylonian and Egyptian mathematics around 1900 to 1800 BCE, more than a thousand years before Pythagoras lived. Early practical and written evidence from some Babylon (c. 1800–1600 BCE) clay tablets, most famously the Plimpton 322 tablet, list advanced Pythagorean triples (sets of whole numbers that fit the formula showing a clear numerical understanding of the relationship)
After collecting my newspaper from the centre of town, my son picked me up and we proceeded to Kidderminster in order to participate in our day out on the Severn Valley railway. The Severn Valley Railway is a 16-mile (26 km) heritage line running through Shropshire and Worcestershire between Kidderminster and Bridgnorth and is one of the longest preserved railway lines in the country. As speeds on heritage and preserved railway lined are restricted to 25 mph, the journey from one end of the line to another takes the best part of an hour. My son had paid for my ticket so as part of the deal, I was going to buy all of the food for the journey so we treated ourselves to a traditional cooked English breakfast at Kidderminster station. The weather was glorious and the countryside appeared to have made a good recovery after the droughts of the summer and we enjoyed greatly our little trundle through the Worcestershire and then Shropshire country sides. At Bridgnorth, a traditional English pub is built int one end of the station platforms so we enjoyed a pint of mild beer here and then some sandwiches before we made a leisurely trip back again. The train was busy but not overcrowded and my son and I had a good opportunity to tell each other abut all of our current ventures. This evening, I may well be going out again because the widow of one of the prominent parishioners in my local church has let it be known that she would like to commemorate the year’s anniversary of her husband’s death with tea and biscuits after the normal evening services so I may well go along to that and see who I happen to meet. Although I am not a great devotee of football, I notice that England and Spain are playing each other in a friendly match this evening and Spain is always a team watch watching. It has also been announced that after a long investigation, Manchester City who have been amazingly successful over the last few years has been found guilty of a whole series of financial irregularities extending back over the years and the eventual penalties, yet to be announced, are likely to be severe in the extreme.