THE LOGOS OF NUMBER
Logarithms – Class 8 – Main Lesson
The student teacher’s face was a mask of misery as she sat beside the hospital bed – more miserable even than the old-hand high school teacher occupying said bed with a broken leg!
“But I know so little about maths!” Student Teacher protested “How can I teach a 3-week main lesson on – gasp – Logarithms!?” Old Hand stared out the window before answering “Actually you know more than you think, you’ve just never had to focus your knowledge that’s all. After all, you did study maths for 12 years at school. Oh I know, the ideal high school teacher is The Specialist, but circumstances often dictate differently – like now! If you don’t take the unit, no-one will.
It’s true though, few people, in spite of having a fairly sophisticated maths education, use more than simple 4-operations maths in later life. It is truly the province of a select few, like engineers, architects, accountants, surveyors, navigators, and the like. Perhaps your first task is to tell the students when they pop the old “Why do we have to learn this?” that thinking number is as important to self-development, to intellectual and philosophical growth as, say, social science.
The Logarithms main lesson in Class 8 is in the so called ‘spirit’ stream of maths – Numeracy. They’ve already had their ‘soul’ unit, Finance, – and at the beginning of the year their ‘body’ maths main lesson, Geometry. This 3-fold ordering is different in the primary; here the number troika is Mensuration, Numeration and Calculation. Children don’t need to focus on Finance, with its heavy learning towards the occupational world, as do teenagers.”
“So there is a spirit element in Logarithms eh? I thought that it was just learning to use those wretched tables.”
“young people often reject maths because background is lacking in the teaching – spiritual content even! We have to track right back to see how these ‘wretched log tables’ (poor attitude here!) evolved. For a start we have to dispel the myth that maths have evolved from the simple to the complex – the opposite is actually the case.
Mathematical experience for ancient humanity consisted of being immersed in that super-complex Astral realm from which cosmic numerology, and hence all earthly computation, is derived. This is the enigmatic Number Ether!”
“This number ether” continued Old Hand “is a kind of transition zone between heaven and earth, a region of psychic access where the miracle of number-metamorphosis can be actually perceived, metaphysically of course. It is a true Realm of Revelation.
From this infinitely manifold world, our maths processes have degenerated to two simple numbers, with which we can create seeming miracles – 0 and 1, the binary code which is the underlying foundation of all computer calculation – the base two as it’s known.”
“Gosh, in that vein I suppose the discovery of say the Golden Section of the ancients is a greater discovery than the binary – hmmm – 0 and 1, place the 1 over the 0 and you have the symbol of the ‘perfect ratio,’ as the golden mean is called.” Student Teacher was staring at a wall panel in the hospital room, one which just happened to be (just happened?!) the golden ratio – 1:1.616. Perhaps this oh-so-human proportion had unconsciously prompted her little insight – as it does marketers of cigarettes and soap powder who so often construct their packaging in this perfect ratio.
“Perhaps I could begin with this imponderable phenomenon with the class – but where?”
“Why not with the 7 Liberal Arts? These have many laminates of reality – the evolution of the mystery of the Measure of Man, the golden section, being just one of them. In Egypt, where maths came to earth for the first time, there were just four of these ‘Arts’, and all were numerical: Geometry, Astronomy (Time), Arithmetic and Music…”
“Music – numerical?!”
“Of course; anyone knows that all true music is based on mathematical ratios of various kinds; that’s why music is as metamorphic as number; as hard to pin down. The Tone Ether (as in musical tones) and the Number Ether are one, according to Rudolf Steiner, he who provided the insight into this ethereal realm. Anyway, these four Liberal Arts represent physical, etheric, astral and ego respectively – music is, as you know, the Art of the Ego.”
“When I sat in your Shape and Space Geometry main lesson earlier this year, you related to the students that ‘square’ consciousness incarnated with the Egyptians (Chaldeans, Babylonians, it al) who expressed a singular geometric (‘earth measure’) or physical body knowledge. It is they who first infused so much of their architecture and tomb construction – the ‘body’ is the coffin of the Spirit – with the golden section – even their skills in land sectioning has this wisdom.”
“Yes, the knowledge these Egyptians, Hebrews, etc. had descended on a golden path from the sun – a logarithmic spiral path in fact.”
“The word logarithms was coined by the creator of the log tables, John Napier, it means ‘Logos (or Word of Number’. The home of the Logos is the sun. This wonderful logarithmic spiral, found as the basis of virtually all sun-created nature, from snail shells to sunflowers, expresses perfectly the number law behind the golden section – and ultimately the logarithmic tables.
The log spiral’s earthly complement is the circle-based Archimedean spiral, with its even progression. A good exercise to lead your – my! – Class 8 into a receptive mind-frame for logs would be to construct a logarithmic spiral using golden mean rectangles. Oh they might have already done this in primary school, but the high school approach is a little more clinical, based on the perfection of the number principles rather than lots of color.”
“Great! We might even illustrate the golden rectangle as she is found in nature – sans spiral – say in the proportions of the humble sole fish. If logs expressed through the Geometry of the rectangle in Egypt, what about the etheric body – Astronomy?”
“Ah yes – actually some occult commentators regard this ‘star’ body as an astral expression, aster means ‘star’ after all – and arithmetic as the etheric of the four number ‘Arts’. In truth the two are closely braided together, with one living in the other. But we’ll proceed with this designation You see that triangle?”
The soft afternoon sun shone obliquely through the window, the golden light manifesting in invisible air-borne dust. “The right-angle triangle, the one formed by the sun shining through a vertical window onto a horizontal floor, is the basis of etheric Liberal Art consciousness, especially in Greece.
You know that using triangulation, the Greek mathematicians measured all kinds of astronomicalphenomena, like the distance of the sun and moon from the earth – the size of the earth even (these enlightened folk didn’t conceive of a flat earth like the later Christians did). They even worked out – in the spirit of that peculiarly etheric element, time, an accurate day length for the year.
But it was the enigmatic Pythagoras (‘heavenly serpent’ – Zodiac!), with his famous theorem which brought ‘triangle awareness’ to the Measure of Man. Actually the breakthrough was very simple, but the implications profound. Again this number phenomenon is based on the ubiquitous log spiral, with its right triangle as a creation basis.”
Student teacher was staring at the triangle of light as he spoke “The Greeks could be quite concrete you know; someone was playing with counters, simple river pebbles, and discovered the square numbers.”
“The arranging of pebbles in orderly rows of even numbers led to the theorem: a squared + b squared = c squared.” Went on Old Hand without pause “have the class construct this; again they will have done it in a concrete way in primary, but do it now as a purely conceptual construction. Remember, a deeper meaning of this mathematical marvel is that it is a symbol of incarnate Man. So draw it the right way up – with the hypotenuse at the bottom (hypotenuse means ‘stretch below’ – the Greeks knew about the measure of Man!). This way it does not become man standing on her/his head, a negative symbol – a lie!
The two top squares are the head – left and right brain actually, the big one major and the small minor. The right triangle is of course the chest – the right angle itself the larynx, center of ego-speech for the Word-obsessed Greeks. The hypotenuse is, stretched below like our own, the diaphragm – and the large square at the bottom is metabolic/limb.
Oh these Greeks knew so much; like the symbol of the Golden Mean, that which represents the formula: square root 5 + 1 over 2:
Now this is one of the most profound images in number history – or any other kind for that matter, representing in its highest sense, Self in Spirit – Ego in sun. The vertical line is the human being, the ego, surrounded by the protective circle of the Spirit. Oh how this has been corrupted in 2500 years! Most number commentators say that humanity struggled up from a state of log-stupid ignorance in the past to the numerical brilliance which powers our civilization today.
Actually the truth is quite the opposite; with concepts like the living log spiral, a numerological mystery that underlies all created phenomena. Ancient peoples knew, mostly on an instinctual basis to be sure, of this vast number-astral world; one which, as we formalized maths and wrote t down, grew ever more dim for us.
The simplest – and latest! Number concept of all, one which is based on the Archimedean earth spiral, not the sun log, is that which drives the whole computer industry, the binary (two) system of 0 and 1. This is the golden mean symbol separated – the human being has now been removed from the Spirit! The computer is indeed a sunless world – the sun, a soul-less star.”
“…!?! Then it’s our task as teachers, to unite them again – what a responsibility; like constructing a logarithmic spiral, but this time using right triangles…!” Student teacher stopped for a moment, meditating on sun, wall and floor. Slowly the golden right triangle became two, the second larger than the first (square root of 2 to square root 3).
The triangles of light began slowly spinning; soon a luminous log spiral filled the room – no, extended beyond the room – sweeping in its great exponential curve all the way back to the sun even!
“That’s a good idea.” Said Old Hand smiling; he didn’t want his student to get too carried away! “But visiting hours is almost over; to proceed to the third Liberal Art, Arithmetic, that of the astral body, we go to Arabia – to the wonderful Moslem centers of learning, from Baghdad to Damascus to Alexandria. But to find the point where the arithmetical step in our logarithm evolution entered European consciousness, we travel, like the young Leonard Fibonacci, to Algiers, sometime in the 13th Century.
This talented Pisan was accepted by an Arab Number Master, an astral initiate, and taught the old but occult Indian numerical system, that which we essentially use today. These are the numbers 1 to 9. After pilgrimages to Islamic universities all over the Middle East, Leonardo of Pisa returned to Europe. Here he introduced a numeration system with which you could actually calculate – unlike the cumbersome Roma numerals it replaced, which was still base 10 curiously enough. Alas Roman numerals could only be used for counting; as she is practiced in the Natural Number System.
But the most precious gift of all; a symbol of even greater spiritual liberation, was zero – no, not ‘nothing’, Naught – the archetypal etheric Sun Number. This gave unlimited calculative benefits; with this new Integer System, with its number line extending not only forward, but back into minus, one could count any number at all, positive/negative – major/minor. With this treasure, Leonardo had broached yet another of the defenses of the Guardian of the Number Ether!”
“But what about logs? That’s just a history of number – or of the log spiral at least?” complained Student Teacher, hoping for concrete help for his upcoming lesson.
“Ah, but Fibonacci’s name is immortalized in a number series he observed in genetics – the generations of rabbits actually. Besides, it’s hard enough for young teenagers to grasp the principles of logarithms even with this foundation knowledge – let alone without it!”
“Sorry…!”
“Hmmphh – anyway he saw in genetics, and in virtually all living processes, most especially in the orderly growth of leaves on a stem, the laws of the logarithmic spiral.”
“I remember, the Fibonacci Series – 1 1 2 3 5 8 13 21…! Those numbers where the two previous are added to create the third – so that’s a logarithmic progression too eh?!”
“That’s right; with the Fibonacci numbers you can construct a logarithmic spiral with the students numerically– on graph paper. Oh, Leonardo also brought Algebra (Al-kebr) back to Europe in his hand – or should that he ‘head’! – luggage. Every simple sum we do today has an Arabic origin in its construction, even the word for the normal 4-operations calculations – the ‘sums’ – Algorithm, is from the name of a 9th Century Arabic mathematician.”
A crisply starched Sister marched through the ward on rubber soles =- she glanced meaningfully at Student Teacher “We’d better hurry,” said Old Hand in a hurried whisper “now to Music, the 4th numerical Liberal Art. This Ego expression found its apogee in Europe (Arabic music is…well, tinny to our ears). This occurred from about the 16th Century, as the Renaissance, a renewal of ego consciousness, gathered momentum. All great music has a complex sun spiral configuration behind its magic.
John Napier was a Renaissance man – a Scot actually, a fundamentalist Protestant! He was also a ‘man of high degree’ in the Edinburgh Lodge of the Freemasons…and a number genius…a St. John initiate. It was Napier who introduced the golden St. John eagle to the pulpit of every Anglican church – he was indeed an Ego Man. Edinburgh is of course the West or Ego Lodge.
In studying the achievements of those who went before, he was particularly taken with the mystery of what was later called Scientific Notation these are the little power numbers which make 10x10x10x10 simplify to 10 to the power of 4. Oh, how about constructing your next logarithmic spiral on squares?
This sun form is truly remarkable isn’t it? In fact Napier’s Bones, a curiously ego term (the ego manifests in the mineral man, the skeleton), was the first slide rule.
An excellent exercise prior to launching the class on the log tables is to make a simple cardboard slide rule. This way they grasp the mechanics of the tables – and show them a real slide rule too, even if this is now obsolete in calculation circles. Perhaps Napier’s greatest contribution to mathematics, and indeed human thought, was the perfecting of the decimal point; an achievement of the scale of Fibonacci’s zero, or Pythagoras’ theory.”
“Whew! That lot should take up the whole 3 weeks!”
“Not really; this necessary content-rich preparation for the somewhat sterile log-table teaching should only take about 1 week – one day for intro. (to ‘seed the will’ as we call it); then a day each for the number epiphanies we’ve talked about.”
“I suppose I should also show the students where the log spiral appears in nature.” Said Student Teacher with enthusiasm.
“Indeed, there has to be a concrete element to every maths principle, educating as it does right down to the physical body.”
“Ahem!” Sister was hovering around collecting flowers, wiping food trolleys – lights were being turned on and – embarrassingly – Student Teacher was the only visitor left. “That’s a primary school emphasis as I see it.” She said brusquely “We constantly find maths in the world; but it seems to me that, as you’ve described it anyway, it high school it’s maths in man – particularly the ‘men’ (why no great women mathematicians?) who discovered the various number gateways, like Pythagoras, Fibonacci and Napier.”
“Quite so – you can’t deeply understand logarithms without some knowledge of their creator – and those who preceded him. But let’s ship through some of the ‘nitty gritty’ you’ll have to introduce to the students. Some of the obscure but necessary number terms they will have to know are index; indice (to point); exponent (to set forth); base number; standard form; significant figures; power; common log; natural log; characteristic; mantissa; involution and evolution.”
Sister knew that if the young teacher was ever going to leave and let her get on with preparing the ward for dinner, she would have to step up the pace a bit. “And don’t forget to really drum in the Four Logarithmic Axioms:
- am x mn = am+n 2. am ¸ an = am-n 3. (am)n = amn 4. (ab)n = anbn
Of course the secret of logs is that it transforms a complex multiplication to a simple addition – and similarly with division and subtraction. Oh, and construct a logarithmic spiral using log tables…”
“What? Another one?!”
“Sure; another name for the logarithmic spiral is ‘logarithmic equiangular’. This could be a question in the comprehensive test you give the class at the end of the 3 weeks – goodness, it’s dinner time; I’m afraid it’s time to g…”
“Er Sister, do you think we could have another serving for my friend?”
Leave a Reply