As a schoolboy, struggling with mathematics, the name Pythagoras struck terror in me. I remember staring at the hated formula below and thinking I’d never get it…
In non-gobbledegook, the equation reads: (a squared equals b squared plus c squared). I can hear the teacher’s voice now, confident that everyone would find it intuitive!
Outside, the summer was passing, yet there we were in a hot classroom with dry as dust letters that could also be numbers… And not just that – not that the numbers themselves weren’t bad enough – we had to ‘square’ them as well! What sort of torture was that?
The language of mathematics eventually became a friend, but not before I had to talk myself down from the night-terrors of squares and equations. So, as a prelude to creating some unusual and powerful breathing to go with last weeks’ ‘elements’ exercise, let me share some of the insights about the inner work of Pythagoras, one of the greatest scientist/philosophers the world has ever known.
Years after that childhood terror, and as competent with maths as needed for a career in computing, I came across the diagram below, and realised there was a much better way to teach this stuff…especially if you had a philosophical leaning and wanted to understand the inner meaning of all numbers – of the key to the very idea of quantity, itself.
The Greeks were wonderfully literal in their descriptions. They knew that when you multiplied a number by itself, in this case, ‘a squared’, it also described the AREA marked out by two lines of equal length (the boxes above), set against each other at a ‘right angle’; for example, box a, above, times itself, or a-squared. That square would have an internal space – an area – of one line length times the other. In this case, they are both the same number, so the result is that number multiplied by itself – or turned into a SQUARE.
If you contemplate the properties of the above diagram, you can see the clear linking of the square and the right-angled triangle.
Pythagoras was fascinated by triangles, seeing that many things in nature had two different aspects that were resolved by a third connecting them. In this way, the world moved forward, harmoniously. His most famous triangle is below.
The elements are as follows:
1. It has three sides, and three angles, hence it is a ‘tri-angle’. Ignore the large numbers in the diagram, for now. Their significance will emerge, later.
2. Two of the sides join in a special angle of 90 degrees. This is the same angle as that within a square, in fact, it is the only angle in a square. A square is a very special figure, as we shall see, later. The little square figure indicates that this triangle’s core angle is 90 degrees, otherwise known as a Right Angle.
3. There are three sides to this triangle. The longest side is always opposite the square figure that indicates the Right Angle. The longest side opposite the Right Angle is called the Hypotenuse, which originally meant ‘that which stretches under”.
4. Something that ‘stretches under’ or ‘runs beneath’, like a root on a plant, is a foundation that supports the rest of the structure. In the case of the right-angled triangle, the Hypotenuse of the triangle – that which unites everything, is reflected from the square sign opposite. Neither can exist without the other. The square sign – the right angle – has no dimensions. It is a fixed ‘understanding’ of squareness that is the basis of a unique relationship between two lines. The square is found throughout the universe. Most of the time it is invisible.
5. The square is also the basis of the dimensions of physics and mathematics. A point has no dimensions, just a theoretical position. The line has one dimension, which is length, this is the first use of numbers and direction. The parts of the line have to maintain a consistent direction or it’s not a straight line. ‘Straightness’ becomes foundational, like our square, in everything that follows. Straightness is an extension of Square as an underlying principle.
6. Beyond the straight line, which could go on forever and get dull, there is a need for creation to become more sophisticated in its unfolding of ‘form’. The combination of a square angle (90 deg) and another straight line defines the next dimension, that of an area, generally known as a ‘surface’. The surface is continuous across two dimensions, it’s no longer just a line, and it has an area, whose dimensions are the multiple of each line. A triangle is a surface, the simplest of surfaces, and in its architecture we can see all the principles of creation, plus one more: the two extensions from the original point, created by the ‘square’ or right-angle are ‘resolved’ or ‘made useful’ by the hypotenuse, that which stretches under, or joins, connects, unites, limits.
The process of the creation of form, on which all else in our material world is based, is therefore seen to contain an ongoing inner process, the reflection from the origin (the original square) to the limit of the extensions, in the form of the link between the original invisible square as right-angle and the largest side of the triangle.
In next week’s post, we will continue this foray into the mind and work of Pythagoras, and the further implications of his work. Before closing, however, it’s interesting to reconsider the most famous of the Pythagorean triangles, below, in the light of the above and the following questions:
Q1: can you locate the origin, the primary square, the first length, the second length and that which ‘stretches beneath’, linking the whole creation back to the origin?
Q2: Can you translate the Pythagorean equation below into its ‘action’ in the world, in line with the ‘creation story’ above?
(As a side note, a triangle, explained in this way, has sometimes been compared to the symbol of the Bow of the Archer…)
In the closing post of this ‘Intention’ series, we will consolidate the answers to the above into a single breathing exercise to add to the journey of the elements in Part 2.
Other posts in this series:
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©Stephen Tanham 2021