**CIRCLES AND CYCLES
**Problem-solving – Class 7 – Middle Lesson

The very first number concept presented to Class 1 children on the first day of their first maths main lesson – Counting – was the Circle. From this eternal symbol of Unity, all other created phenomena derive, whether numerical or otherwise.

This lovely but oh-so-simple form, expressed as zero, embraces universal spirituality, being the form of the heavens themselves. But being, as a sphere, the form of the earth as well, of the Physical Body, it is rightly an expression of the Body as well – of the Will.

The circle/sphere was the first created principle – Ancient Saturn – according to the Akashic-accessing Rudolf Steiner; so in an apparent contradiction, it represents both naught and one – the former its spiritual life, the latter as manifest on earth.

Almost all of the 42 maths middle and main lessons throughout the 7 years of primary school – of which this is the last – began with the circle, or Unity; they then proceeded through from duality to plurality – and concluded with Unity again. This 42 is actually the number in occult numerology when the human being passes from a sun-blessed stage to one of ‘spirit’, beginning with Spirit Self.

This can account for the so-called mid-life crisis which descents on many people at around 42. It is also the number of the generations; after the 42^{nd} there is no genetic element left in the human being of that far-off ancestor. (There are 42 generations from Abraham to Jesus). Again we enter a purely spiritual phase, free from ancient inherited ballast.

Unity to Unity through the 42 maths units is the way of life; the way of the number mysteries; the way of the Spirit; and the way of wholesome education. It there an equivalent mid-life crisis in those number units to follow in high school.

In some of those primary units the sun would have been used as an image of Unity, or holism, this is a spiritual sphere of a different kind. Helios, the Greek Sun god, fathered a daughter, Circe, the circle had come to earth. As such she was depicted as a sorceress – a ‘ring maid’. When spiritual principles become matter-bonded, they can be used for good or ill. The circle, as a wheel, can certainly serve humankind, or it can be used for reprehensible ends. So who would have thought that an abstract idea like the circle would have a moral dimension?

If maths (meaning ‘to learn’) is not taught from an ethical platform, it were better it not be taught at all. In fact even ‘earning’ itself has its moral obligations; what one does with what one learns either benefits the world or helps destroy it. So this Circles and Cycles (a moving circle) completes the circle, so to speak, of the child’s number learning in primary school.

In life, curious to say, the number 42 represents the *completion* of the Sun Cycle, as Rudolf Steiner informs us. We are moon beings for the first 7 years of life; from 7 to 14 it is Mercury; 14 to 21, Venus – and from 21 to 4 2 – three 7-year phases – we are Sun beings.

In this 21-year period we are supposed to develop our Sentient, Rational and Consciousness Souls. After 42 we travel the lonely outer-planet path of higher spiritual potential – or we’re supposed to at least! This 42-unit primary cycle also works with Language; significantly the only other subject that it does.

How appropriate then to conclude where we began; and there’s such a lot of interesting circle-cycle content to find as one ranges freely through the manifold halls of the mansion of mathematics. The mighty edifice dwells safely in the spiritual jurisdiction of Taurus – a perennial inspiring source if the teacher would but throw open the psychic doors and let it in!

The consonant, of The Spiritual Twelve ascribed to Taurus ins ‘R’, the only air sound, or aspirant, in the alphabet. R is formed when the breath *rolls* out onto the ether in a trilling series of cycles, or strings of circles. This is not surprising really, as the ‘Quality’ of Taurus, again nominated by the Doctor, is *will*. Will of course always manifest in circular (cycular!) or wheel forms. So what does all that have to do with this unit? Well the spiritual basis for any lesson forms the backdrop of truth which children somehow intuit is there – or not.

The following is a sampling of the seeming endless banquet of circle content; one which shouldn’t be regarded as definitive, or prescriptive even, merely as pointers to what is appropriate to the spirit of this 3-week middle lesson, and to the age of the child.

Many ‘rolling’ words begin, or contain, the Taurus consonant, one being ‘rotation’. The mathematics of rotation introduce the Complex Number System to the child. This is where all points on a plane can be determined where a body, theoretically or actually, rotates through space – and Space Maths is where this calculative aid is especially useful. Back on earth, electrical physics for example that which deals with the movement of electricity through a magnetic coil, employs circle maths. It seems Circe has the power to cast a spell in the electrical field as well!

A rotation of 360° is represented by 1; 180° by -1; therefore 2 rotations of 180° are (-1) x (-1) = 1 = 360°. But what of a rotation of 90°? Whatever the number, when it is squared should it be -1 or 180°? No real number squared give -1. Hence the formula square root of -1 = I; so a 90° rotation = i: i x i = 1 !

It is this tiny letter i which gives us the freedom to plot the impossible on a plane diagram. Every complex number is written as a real number plus i times another real number. For instance a rotation of 45° is ½ square root of 2 + i ½ square toot of 2. Now take a deep breath; when we multiply this number by itself, we get i; 90°. Two rotations of 45° make one of 90°. This becomes clear on the plane diagram:

The *real *numbers are the points on the horizontal and vertical lines; the *complex* numbers, those on the plane. The horizontal line is the ‘axis of real numbers’; turn this 90° clockwise and we have the ‘imaginary numbers’; or real numbers times i. The vertical line is called the ‘axis of imaginaries’ – very imaginative! The complex number is found on the grid coordinates by adding a real and an imaginary number.

Another series is circle problem-solving; after all, this unit is the *Problem Solving*, the third of ‘thinking’ strand of the Numeracy Stream. This particular problem is based on the clock, and the continuous changing angles created by the movement around the circle of the two hands. The amount of rotation around the circle is the *angle*; in the wonderful sexagesimal system introduced – or intuited – by the Arab scholars of Baghdad, the Angle of Unity, of the whole circle, is 360°.

In rotational geometry, each degree is divided into 60 minutes, which give 21,600 minutes in a circle. Each minute has a further 60 seconds, which make 1,296,000 seconds in a circle. In *time *measurement however, there are 60 minutes in an hour (one full rotation of the minute hand, 360 seconds in an hour, and 43,200 seconds in one revolution of the hour hand, or 12 hours.

In rotational geometry there are 720 minutes in a full rotation of the hour hand (12 hours x 60 minutes), and 43,200 seconds – to provide even more ‘minutae’ to this section! Of course all these numbers – these true ‘Circe’ numbers – factorize to the nth degree, so to speak. Using a protractor on a clockface, with 0° on 12, we find 30° at the 1; 90° at 3; 180° at 6 and so on.

What will the angle be – here comes the problem-solving – at half past two? The minute hand is on 6, the hour hand halfway between 2 and 3, with the resultant angle 105°, or so the protractor says! Even non-mathematical children love this ‘hands on’ problem-solving.

Problem 2, solved mathematically this time, not mechanically; What is the first time, after 12 o’clock, that the two hands are together again? The hour hand moves at 5 paces an hour, the minute hand at 60 – the difference is 55.

The gap between them widens at 55 spaces an hour, so the gap becomes a full lap after 60 fifty fifths of an hour, or 1 and one eleventh hours. This is one eleventh x 60 minutes = 5 and five eleventh minutes after 1 o’clock – when is the second time they’re together?!

Naturally the vocabulary of the circle or cycles should be revised, including such old favorites as: angles, minutes, seconds, radius, diameter, tangent, center, perpendicular – and the various stable formula; 2 Pi r, Pi r squared, Pi d, etc.

Talking about the perpendicular, this is used in measuring cycles, or moving circles. Everything in the world, from the hypothetical atom to a planet singing along its elliptical orbit, has a specific vibration or frequency – or to use a more poetic term, ‘song’. This is in the form of *waves* which move through the air; they are not the air, but an etheric pulsation through it.

These cycles all have their genesis in the wheel or circle; simple waves are like the standard ocean model, which complicated ones are made up of several of these simple waves combined. A pencil tied to the rim of a bicycle wheel (P) as an extension of one of the spokes will scribe, when the wheel is turned, on a sheet of paper a wave from (beginning at Q on the horizontal) with the width and height of the wave the same measurement – a = b.

The hub of the wheel (O) continues to move along the horizontal center line. Each full wave is the 360° turn of the wheels, with a 90° turn at the top – at 180° it is at 3 o’clock – 270° at the bottom – and 360° back at the top again! In all but 360° and 180°, the height of P above or below the center line can be described as a fraction of the length of the spoke.

We find this by dividing height of P by length of spoke. When P is above the center, we represent its height by a positive number, when below a negative. The size of the fraction is determined by the angle the wheel has turned.

If the wheel turn 30°, to 10 o’clock, this means that the vertical line from rim to horizontal is half of spoke length – a over b = ½, the wave is half as deep as it is long. A pencil fixed to a spoke at this vertical/horizontal intersection point will scribe just such a wave. These are the same wave forms which appear on an oscilloscope screen; a machine for visually displaying the variation in electrical currents.

This wave phenomenon is implicit in sound mathematics; when describing sound cycles, with their perpendicular frequencies from, say in the scale of C, 256 to 512, here we find the white keys on the piano. To build up a scale that starts with another tonic, or *keynote*, instead of C, we use the black keys as well.

For this we can make a simple scale-fining *wheel*; turn the small wheel until the 1 lines next to the tone you want as your tonic, then the numbers 2 to 7 point out the other tones in the scale – in the correct order. In truth only sound moves in waves, or frequencies, with light traveling in rays, or straight lines.

And a last, perhaps easier one, having a ready-made formula at our disposal – S = ½ gt squared. This was supplied by the gravity-loving Isaac Newton, and tells us the distance a flying object, like an ejected cannon ball, will drop to earth – its ‘trajectory’.

S is the distance, g is gravity, which is 32 feet per second for every second; e.g. the ball falls 32 feet in the 2st second, 64 the second and so on. In theory, a cannon ball, if gravity held good at infinite heights, could plummet right through the earth if dropped from a great enough height!

This is the ‘g’ force – t is the time the cannon ball stays in the air. This equation can be transposed to find either g or t.

So what has this done with the circle? Not much I suppose; apart from the fact that a spherical cannon ball is traveling a circular (actually parabolic) path around a globular earth. But the primary maths curriculum has to stop somewhere, as we depart into infinity and eternity from our splendid spherical home.

Even these two mysteries are circular, or wheels, in the great ordering of the universe; with infinity relating to space less space, and eternity to timeless time. However our so-called infinite universe is thought of by cultures everywhere as being a great starry dome, and endless time as, according to E = MC squared, a greater than great circle. Both concepts were woven like golden thread through the first numbers lesson in Class 1, Counting, and conclude here, after 42 wonderful units at the same place and time – circles n cycles indeed!

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