So far, we have examined how Arthur M. Young, inventor of the Bell helicopter, engineer and astrologer/philosopher, used his skills and insight into how our minds determine meaning. Within this, he began to discover that there was a graphical symmetry to this process; a set of shapes that explained many of the ancient symbols that mankind has come to view as sacred. These will shortly be unveiled in more detail, but, first, we need to complete our tour of the foundations of how he approached it, for the symmetry emerges from those foundations and how we represent them.

In the last post, we looked at how Isaac Newton investigated the motion of things that move, discovering that – for example in the motion of a cannon ball – there were different aspects, faces, of that motion; and that although they were often hidden, they were tightly related to each other. Arthur Young used the equations that Newton produced for this. Unfortunately, this led us into numbers, squared numbers and, and horrors, cubed numbers! Several brave readers made it to the end of last week’s post, but not without difficulty. So, for this week, I decided to take a small detour to illustrate how these types of numbers can be see as pictures instead of fear-inducing maths.

As a child, I had a terror of maths, assisted by an ex military ‘Desert Rat’ of a headmaster who believed that beating boys and throwing board-dusters at girls would help their education. That was the 1960s, not Victorian England; and the dubious joys of a Church of England country primary school. Times have changed, but the horror of seeing something squared or cubed has not. So, by way a small gift, let me share with you one of the most beautiful insights I ever learned – though, sadly, beyond my school days.

It was the ancient Greeks who developed the idea of squares and cubes and the numbers that represented them. They ‘saw’ numbers as representing both qualities and quantities including what they thought of as other things, like distance from a point of origin.

In the diagram above, a unit of distance, marked ‘1’, (inches, metres, feet, etc) is added to others, in the form: 1+1+1=3. Nothing too complicated about that; it’s simply addition, the sort of thing we use every day.

Now, imagine that these numbers are a child’s counting blocks, as above. We arrange them in a line to produce the three, again. But this time, we begin another line of them with the last block of the first line. In doing this, we have changed the nature of what lies before us – what we are creating. As an example we might say we have begun to make a picture frame to contain our favourite photograph. In the process (and intuitively to our minds) we have turned a ‘perfect’ corner to begin the second row of blocks. This perfect corner is what we all know as a ‘right angle’, so named because of its special – and ancient – properties of ‘rightness’.

We can fill in our photograph frame with other blocks. Because of the right angle – which we know to be ninety degrees – the block will all fit together to form something dramatically new. What started off as line has now become an area…. Our simple maths formula was just 1+1+1=3. But now, we have an area whose properties can be derived from the counting blocks that make each side. We have a choice: we can simply count all the ‘one’ blocks, or we can ask our Greek teachers if there is a quicker way. They will tell us that we can multiple or ‘times’ the length of one side by another. This would result in 3 x 3 = 9. Again that’s not too frightening. Our picture frame could have been a 3 x 4 rectangle, which would have given us an area of 3 x 4 = 12.

The first one above (3 x 3) has a special symmetry in that each side is the same length.  Because of this identical symmetry, our line of three has become not just an area of nine but a SQUARE. This is the origin of square numbers: they are the same number multiplied by itself. And they produce a very magical figure – the square. To the ancient Greeks, this was very special. They envisaged that the square reflected a manifestation of divinity. From an origin – which had no quantity, but it had a location – it led to a line, which did have a dimension, then to another line at the ‘right’ angle to produce a square.

You can’t square a number to get a rectangle; you can only get a square. Anything ‘squared’ therefore is based upon the union of two identical things, but arranged in a certain way, so that they have a relationship to each other. In this case that relationship is ‘times’ or multiplication. We shall see later in this series of blogs how Arthur M. Young expanded these relationships to provide us with a full diagram of human meaning – and reconciled much of the diverse ancient wisdom in the process.

Back to our squares and rectangles. A rectangle is useful, of course – most pictures are rectangles – but a square is ‘perfect’ and quite capable of being used as a sacred symbol, as, for example. Masonic teaching shows. Within the Masonic teachings (I am not a Mason, but have great respect for what masonry sets out to do) someone of right character is described as ‘being on the square’.

Let’s  summarise to far:

We have an invisible point of origin (where we begin our construction or drawing);

As soon as we start to draw our line, we have a point, which has no length, but exists;

When we have an extension to that point in a certain direction, we have a line: in this case of length three units – but this could be any number.

When our length (or extension) is done, we turn our construction through 90 degrees – a right angle – and begin another line (effectively from another origin, but at a different point and connected with the first).

We could have continued this process, just doing the edge of our picture frame, and we would have arrived back at our start point – having created only the edge of our square. But along the way, we learned that to ‘square’ the length gave us the area contained by the whole figure: a surface or ‘plane’ of a higher order.

Can we continue this, or is the process finished with the area of our picture frame? We learned that the mystical key to the creation of a higher order was the Right Angle – 90 degrees. This whole process has been about the generation of space in which life (and motion) can happen. Can we take our figure and extend it through another 90 degrees, without repeating what we have done? And, if we get there, what will it teach us about a number cubed?

The picture below contains the answer. Enough for one post, I think. We will elaborate on this next Thurday…

To be continued…

{Note to the reader: These posts are not about maths or physics; they are about a unique perspective on universal meaning created by Arthur M. Young. If you can grasp the concepts in this blog, your understanding of what follows will be deeper.}

Previous posts in this series:

Part One,   Part Two,   Part Three,  Part Four

©️Stephen Tanham

Stephen Tanham is a director of the Silent Eye School of Consciousness, a not-for-profit organisation that helps people find a personal path to a deeper place within their internal and external lives.

The Silent Eye provides home-based, practical courses which are low-cost and personally supervised. The course materials and corresponding supervision are provided month by month without further commitment.

Steve’s personal blog, Sun in Gemini, is at stevetanham.wordpress.com.

You’ll find friends, poetry, literature and photography there…and some great guest posts on related topics.

Everything is in motion… Arthur M. Young and Isaac Newton both knew that, but in different ages and different ways. Let’s take a slight detour into some basic ways of looking at one of our fundamentals – the way things move. Our search for Arthur M. Young’s ‘geometry of meaning’ will be enhanced if we can enrich our vocabulary…

Someone in the age of Newton would have said. “This chair upon which I sit is plainly still.”

We can be cleverer than that, now. We all know that our planet is rotating once per day. We may remember that the Earth orbits around its sun once per year. We can even know that the atoms from which the chair is made are themselves in constant motion, albeit within a quantum envelope which renders them solid only when they are observed. The chair is therefore in constant motion, but most of that motion is irrelevant to the scale of human life. The rotation of the Earth is not likely to upset the stability of the chair, but it would be theoretically possible to create a hyper-sensitive chair that was…

Newton did not know of atoms, though the ancient Greeks discussed their necessity. But he knew that there had to be a limit to how many times you could divide something. At that limit you would find the essence of matter. He was very adept at envisioning the practical consequences of pursuing things to their limit…

He knew that things moved differently; not just in how one thing could overtake another, but that – within how they moved – there were differences of what we now call ‘rates’. To grasp this, we need to revisit the idea of a rate. If I have a dripping tap, and it results in one gallon of wasted water, measured over an hour, then I have loss of one gallon of water per hour. That is a rate: it is one relevant number divided by another – something per something else. It is a measure of how something that changes (dynamic) behaves with respect to something else. But our dripping tap may not waste water in a uniform way. Within that hour there may be peaks and troughs in leakage due to aspects or factors not known about in our ‘averaged’ one hour period. This is important to hold in mind when thinking about ‘motion’, too.

In Newton’s time, it was known that the ‘motion’ of things had different aspects. Imagine Isaac Newton as a child playing a game whereby he used a fallen branch of a tree, suitably trimmed with his penknife, to strike stones in his garden to see how far they would fly. He would notice that such stones went from being stationary (at rest) to suddenly going as fast as they might (a maximum) before travelling through the air in an arc and falling to earth again. The motion of the stone would therefore vary from nothing (taking out the Earth’s motion) to maximum speed – as it climbed into the air; to a point where what we now call gravity caused its upward motion to cease and its downward motion to increase, even though it was still moving away in terms of distance from the child Newton in the garden. Thereafter, the grass and earth would tangle its motion and it would come to rest again.

If we measure the whole of this motion, we might simply conclude that the stone was whacked by the strong child wielding a stick and shot down the garden for a length (distance) of, say, 10 metres. If a modern time instrument had been available, we might also discover that it took five seconds to come to rest. This would be accurate as an ‘average’ of what had happened, but would tell us little of the stages of the lifecycle of that overall motion – the interesting bits!

The above motion of the stone (with the help of a modern timer) would yield a measure called the speed or velocity of the stone of as: 10/5 = 2 metres per second: distance divided by time. But that’s not what happened, except seen as a historical thing. What really happened is that when child Newton whacked the stone, it didn’t just have a constant speed; its speed changed from nothing to its maximum value, sufficient to propel it (with the correct angle of strike) into the air in its graceful, if short, arc. Thereafter it slowed and sank through the air while still travelling along the line of its trajectory – the direction in which it was whacked. After this, it landed, bounced and came to rest in a scruffy (but real) way in the tangle of grass and mud.

Aside from my borrowing of his childhood, the real Newton had the genius to realise that the first part of the motion, (from rest to its maximum) was not just speed, but an increase of speed (from nothing to its maximum) that had a different rate. This was caused by the whacking of the stout stick, which transferred its energy to the stone, slowing the stick and thrusting the stone into space. This change of speed or velocity was named acceleration, and it was seen by Newton as something different to velocity, itself. This was a breakthrough in thought and measurement, and marked Newton as a true genius. It would take hundreds of years for Newton’s discoveries to filter into the mindset of the age. Many people today have little idea what he achieved, and yet our age of powered motion is built on his discoveries and the accompanying mathematics of calculus. The “Newtonian” world is the world of classical physics, and this view of how the world operated persisted until the advent of Quantum Theory in the early years of the last century.

Returning to Arthur Young’s discoveries. Young examined the symmetry of what Newton had discovered in the following way.:

Motion begins with distance from a start-point. In our example above the stone travelled ten metres. This is simply a length, which we can call ‘L’. A length ‘L’ applied to a start point (or Origin), without consideration of its motion, simply gives us a new position.

If we want to go further and investigate the real motion of our stone, we consider the time it took to travel the distance. We can call this ‘T’. The length (L) per time (T), written L/T (length divided by time) gives us a rate called speed or velocity – example miles per hour. This ratio of L/T is a basis for all motion and reduces things to their simplest expression.

So, what about acceleration? Remember that this is an increase of velocity not distance. If my car accelerates, it is now travelling at, say, sixty miles per hour rather than fifty. The acceleration has been ten miles per hour, per hour. In other words the rate of change of the velocity.

Summarising this:

Position = L

Velocity (speed) = is the rate of change of position or distance = L/T

Acceleration is the rate of change of velocity, which is L divided by T times T. This new expression, T times T is written T squared, T with a little ‘2’ to the right of it like this: T²

Arthur Young was pursuing the fit of the science of motion to the Fourfold model of meaning we discussed in the first three of these blogs. He needed a fourth term to follow the sequence:

Length (L),

Rate of change of Length, (L/T or velocity)

Rate of change of rate of change of Length, (L/T² or acceleration)

The missing term (L/T³) would be the next in the series and would complete the integration of the human world of motion with Young’s fourfold map of universal meaning…

But there was no recognition of a fourth term (L/T³) of Length and Time in physics… Yet Arthur M. Young, creator of the modern helicopter, knew there was a commonly understood concept that matched this – he had used it to make his helicopters safe…

To be continued…

{Note to the reader: These posts are not about maths or physics; they are about a unique perspective on universal meaning created by Arthur M. Young. If you can grasp the concepts in this blog, your understanding of what follows will be deeper.}

Previous posts in this series:

Part One,   Part Two,   Part Three,

©️Stephen Tanham

Stephen Tanham is a director of the Silent Eye School of Consciousness, a not-for-profit organisation that helps people find a personal path to a deeper place within their internal and external lives.

The Silent Eye provides home-based, practical courses which are low-cost and personally supervised. The course materials and corresponding supervision are provided month by month without further commitment.

Steve’s personal blog, Sun in Gemini, is at stevetanham.wordpress.com.

You’ll find friends, poetry, literature and photography there…and some great guest posts on related topics.