Watch My Line

“The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.” – Pythagoras

At the equivalent of High School which I attended in London, we all had to take basic Mathematics, consisting of Arithmetic, Algebra, and Geometry. I could do them all, and more or less understand them, but somehow Geometry seemed to make much more sense than the others. Geometry (which means measuring the Earth) had to do with points and lines, surfaces and shapes. Something about it appealed to me, perhaps because it was more visual.

You started with imagining a “point,” which had no dimensions at all, and so could not be measured. And it could be anywhere. It was a “nothing.” Then you imagined that point moving, and leaving a trail, called a “line.” If that trail marked the shortest distance between its beginning and its end, the line was called “straight.” That line had no thickness, but it did have length, which – aha! – now gave us something to measure.

From there, you could bend the line on a flat surface to make enclosed shapes, like triangles, squares, and circles. (Circles were particularly interesting, because they were totally un-straight, but they gave you straight lines to the center [radius] or all the way across [diameter]). And you could then jump off the surface into the mysterious Third Dimension, making things like cubes and spheres.

It was a beautiful system – but what genius created it, and what was it good for? Credit is usually given to a semi-mythical ancient Greek named Euclid, who wrote a book called “Elements,” which apparently was so good that it was used in schools almost up to our own time. The Greek connection explains the use of many Greek terms like “isosceles,” “ellipse” and “hypotenuse,” which you hardly ever come across outside of Geometry.

Speaking of hypotenuse, there were of course many other magnificent minds involved in this whole process, and one of them, Pythagoras, is responsible for a theorem in which that word occurs, and which may be the only tit-bit many of us remember from our encounter with geometry.

This famous theorem has to do with triangles – but only with one special kind of triangle – the kind in which one corner is perfectly “square” that is, it has an angle called a “right” angle. Angles are measured in degrees, out of a possible total of 360. (The fact that this is so similar to the number of days in a year must be more than co-incidental.) And a right angle is a quarter of that, or 90 degrees.

In this theorem, the hypotenuse is the line of the triangle facing the 90-degree corner. And what makes this all so important and useful is that, if you know the length of any two sides, you can calculate the length of the third side. This is the basis of that whole wonderful system called “triangulation” which surveyors are still using to measure the earth.

The whole idea is based on making the sides of the triangle into squares, and showing that the two smaller squares, when added together, have the same area as the large one i.e. the square of the hypotenuse. This seems sensible – but Pythagoras was able to prove it. And what also made geometry fascinating to me was that, starting with what you know, you could logically prove, in a series of steps, something you didn’t know. When you get to that conclusion, in a geometrical theorem, you are traditionally entitled to write, “Q.E.D.” This stands for the Latin, “Quod Erat Demonstrandum,” meaning “Which was to be proven.” But at school, we were jokingly told that it stood for “Quite Easily Done.” We were also told – and I don’t know how much truth there was in this – that, in their training, police detectives had to study geometry, because it provided good exercise in logical thinking.

In any case, it was this same scrap of knowledge which Gilbert and Sullivan included in one of their most famous songs, “I Am the Very Model of a Modern Major General,” (from The Pirates of Penzance). In this aria, Major General Stanley boasts of the wide range of his knowledge in all matters (except military ones). And among his intellectual accomplishments, he proclaims that:

“About binomial theorems I am teeming with a lot o’ news,

With many cheerful facts about the square of the hypotenuse.”