EUCLID AND THE PHARAOH
Geometry – Class 7 – Main Lesson
The Room, or should I say the Grand Hall, where Euclid toiled was salubrious indeed, being part of tan Academy devoted to scholarly pursuits in the pharaonic palace of Ptolemy I at Alexandria in northern Egypt. The Pharaoh had been watching his busy countryman (both Ptolemy and Euclid were Greek) struggling with, as it later turned out, a brilliant solution to the construction of a pentagon.
“Dear me,” said Ptolemy, impressed by the Math Master’s perseverance, if nothing else! “that looks terribly difficult; can’t you show me – I mean me – an easier way?”
Euclid turned to his friend, a thin line of scorn on his lips “Alas there is no royal road to geometry.” He said smoothly. For this, lesser men would have been fed to the crocodiles; but the climate in Egypt in 3rd Century B.C. Egypt for the revelation of the Number Mysteries was actually more clement than in Greece – that’s why Euclid migrated.
Greece, in this period, was enjoying the cultural beneficence rather of The Word, or Logo, swelling in the vernal sign of Aries as it was – Sense of Word. In the strict sense so of course was Egypt; which is probably why they welcomed the Greek Ptolemaic Dynasty.
But cultural canons die slowly, and after 400 years of the transition from Taurus to Aries, the center of the Bull culture, Egypt, was still largely living in this temple-based number revelation.
Taurus is, of the 12 Philosophical Standpoints given by Rudolf Steiner, that of Rationalism. That’s why the Egyptians (and other of that era, like the Babylonians, Hebrews, etc.) saw their world in terms of reason; in the extreme, of a frozen nature to be sure. The numerically perfect pyramids are an example of this (not much spontaneity here!), as is that opus of wonderful Kabbalistic Numerology grudgingly given to the world by the Hebrew esoteric masters.
Number has its home in the Astral world; the Number ether as Rudolf Steiner calls one aspect of it. Taurus is of course the Astral cardinal point in the Zodiac. This Taurian culture actually mummified great black bulls – the mighty Apis – so that the animals’ bodies would not corrupt. The Number Adepts in the bowels of the Temple would, over time, use these souls in limbo as an information center for higher worlds. In the 3rd post-Atlantean – Taurus – civilization the bull was thus revered and spiritually exploited.
The Greek equivalent of Taurus, the wicked virgin-eating Minotaur, shows how far the Bull had fallen from grace in a Sky Ram culture! This anti-number ambience (look what happened to Diogenes!) was what Euclid was escaping from.
Of the three main streams of mathematics, Mensuration, Numeration, and Calculation, Geometry is the first; of ‘measuring’. This calls on the forces of, above all, the Will of the number adept or school student. One really has to become physically active to measure something – a valuable hint to a teacher presenting Euclidean Geometry to his/her Class 7.
As well as understanding the seemingly limitless theorems, the children should go out – or in – and measure things. In the case of Euclidean Geometry, angles mostly. Angle is, you might have noticed, a curiously similar word to angel; although the first means bent, the second, messenger! A ‘bent’ person, in a modern sociological idiom, is someone who is on the cutting edge of consciousness!
Anyway, at what angle does that palm frond leave the stem? What height is the equilateral triangle in the latest ‘crop circle’ impression left in the school playground last night!? If the Euclidean measurements are devoid of this reality of Life, they remain abstract, creating gravel in the soul rather than jewels of understanding.
Mine you, there is quite a lot of ‘will’ activity in the many drawings, most of them relatively simple, that are required to illustrate the various theorems. As this is still primary school, these constructions should still be done in color, albeit fine and accurate.
Not only does this more artistic presentation look better, but can be a vital factor in describing and comprehending. Say one is dealing with the diagonals of a quadrilateral; how much easier to say:
“The quadrilateral, the blue figure, is a parallelogram because the diagonals, that red cross, have equal length.”
This is far more effective than “Figure a, b, c, d and lines x, y, and u, v…” There can even be light shading of different surfaces, like the sectioned ellipse of a cone. A color element will help keep the interest of these children whose souls travel on a more eccentric astral plane!
Euclid was working away on the definition of the four elements of geometry and their relation to the four bodies of the human being when the Pharaoh, who had been for some time puzzling over a maze of parallel lines intersected by …!! Whatever!; muttered “Why do we need to know this? What can I get out of it?!”
The Mater’s wry reply has become a timeless clip over the thick ear of the philistine “Give the man three pence, since he needs make gain from what he learns.” Ptolemy pretended not to hear!
Euclid knew that to study Geometry was to ‘know thyself; in the deepest possible way. He was an Etheric Initiate, hence his obsession with angles and especially triangles; in fact this was the only area in which he made a truly original contribution to mathematical understanding. The angle-triangle is the archetypal etheric form. Much of his work on Pythagorean number, and other areas, was more a compilation that an innovation.
The gods as usual, were divinely kind in dispatching four Number Masters to bequeath to humankind the ‘whole man’ of mathematics. Archimedes dealt mainly in Mechanics, the ‘physical body’ pursuit – ‘arch’ means first. The physical is the first principle of incarnate man.
Euclid, as mentioned, unveiled the mysteries of the Triangle, the ether body principle. Eratosthenes was an Astronomer, self-evidently assuming that the world was a sphere. He even worked out, along with other cosmic bodies, the plant’s vital statistics. This vision marks him as an astral (‘star’) seer. His name is akin to poetry, art of Spirit self-fully-transformed astral body. Erato is the Muse of Poetry.
Pythagoras described, in sublime numerology, the Ego of man; this was in the amazing insight he developed into the seeming abstraction of the golden mean and other cosmic relationships, especially as they manifested in the natural world.
Python is the fallen dragon, or divine serpent; this is an allegory of the ego descended from its heavenly home to a matter-bound existence – in its capacity for evil, into a sub-earth existence eve. Pythagoras was the music mathematician; music is, according to Rudolf Steiner, the Art of the Ego. The tonic has a frequency of 256, or four cubed; this, to the square root of four, equals four, the number of the ego!
In a subset of this 4-fold ‘bodies’ concept, Euclid saw them expressed in the Four Elements; these are: Point – fire (physical body); Line – air (ether body); Surface – water (astral); and Space is the mineral world in which both global and human ego dwells. These four keys open doors to wonderous worlds; not necessarily only of number.
The fire-physical body can only express as a point; the air-etheric marks off the timeline of existence; and our liquid-astral perception of both inner and outer worlds is based on manifold surfaces. The mineral-ego can only incarnate when space, in the form of a combination of the other three, is made available. All this is to do primarily with our life on earth; geometry does mean ‘earth measure’.
During the 1st Millennium, Euclid’s wonderful legacy, in the form of six books, was guarded by one or other of the ‘Temples’; those which were benevolently conquered by the Moslems.
Thank the good number gods that it was Islam which rescued and preserved Euclid’s Six Books; wherever intolerant Christianity spread, everything considered to be ‘pagan’ (which merely means ‘peasant’) was burnt! It was the inimitable Haroun Al Rachid – a personage thought very highly of by Rudolf Steiner – who, from his Baghdad splendor, first translated the books into Arabic, hence into immortality.
The most important of Euclid’s Great Six was his Elements, which remained the standard text for geometers for the next 2000 years. These fundamentals are still taught in this Class 7 Geometry main lesson today. The books were eventually translated into Latin in the 13th Century.
This reference to Haroun and his Court should be fleshed out a bit in class, being an expression of the ‘Dark Ages’ as it is. In Islamic terms, these were rather the Light Ages, a period of scholarship and culture which in many ways rivalled Ancient Greece in excellence.
In the great recapitulation of the so-called ‘epoch’, 13-year-olds are at the evolving consciousness of that period, from about 300 to 1000 A.D. – The Age of Chivalry. Allusions to Haroun et al will find their stirring complements, not only in the souls of the children, but in the World History and other lessons taught in this marvelous transitional year – transitional from primary to high that is.
“What are you doing now?” said a curious Ptolemy Soter as he learned precariously over the master’s shoulder. Irritated by yet another royal interruption, Euclid testily replied “I’m constructing a pons asinorum!’
“What?! That triangle doesn’t look like an ‘ass’s bridge’ to me?!” said Soter; Euclid groaned “To prove that the isosceles triangle has congruent base angles is so involved, it’s a metaphor for obstinacy – like trying to force an ass across a bridge!!” Ptolemy wandered off, to inspect a pyramid or something, hoping that a passing slave hadn’t heard this precocious reply.
The teacher of the Euclidean Geometry main lesson won’t have to allude to his/her pupils as donkeys if order is the order of the day; on a solid foundation of angles and triangles, each major geometrical form should be taught separately, and very, very clearly. A good place to begin – as usual – is the circle, where its division into various radii, diagonals, chords and tangents provide a three-ring circus of theorems.
An amusing diversion from ‘straight’ presentation might be to find Pi with a toothpick. Cut one to the width of the floorboards, and drip it, say, 100 times – the more drops the more accurate. Count the number of times it falls across a crack, say 62, – multiply the drop number by 2 – 200 – then divide by ‘crack’ number 200 divided by 60 = 3.2. Well, not quite Pi, 3.14, but pretty close.
From the circle, the physical-fire form, to the triangle etheric-air; again definitions of the different triangles are revised and new theorems taught. When Euclid meditated (all Masters of the arts and sciences did!), whether on the world or man, being a Mathematist he saw only numbers. His vision of the ether body (larger then than now, being almost the same size as the physical today) comprised a complex pattern of radiations. These created angles of incidence – of invisible light – of every possible configuration; something like the arcing World War II searchlight over London.
From these he arrived at this ’94 Propositions’. Not that all of these have to be taught in Class 7, but a selection of the most important ones at least; like the base angles of an isosceles triangle are congruent. (Always use and teach the correct terms; congruent rather than equal).
From the circles and triangles, move to the Quadrilaterals, with their emphasis on lines reaching to infinity. A standard here is the ‘parallel axiom’; in a straight line intersecting two other straight lines, if the two angles on the same side of the line are less than 90°, then the two lines will meet – somewhere!
Even though this exposition of theorems seems innumerable, the geometer always tries to simplify to a set of comprehensive laws. Euclid devised 10 of these, his 5 Axioms – and 5 Common Notions; which he collectively called The 10 Postulates.
The first axiom, to give you the idea (these are easily found in any decent reference work on Euclid) is – ‘Given 2 points, there is an interval between them.’. Sounds self-evident doesn’t it?!
These 10 stand yet again on the certainty of the earth-shaking statements underlying the whole of Euclidean Geometry. ‘If certain elements in a figure are given, then other things other things are given.’ These 10 axioms/notions are of course not arbitrary; Egypt was the civilization of the Ten, with its 10-sign Zodiac, the Decans – and other decimal-mystical connotations. Their Hebrew contemporaries of course had their 10 Commandments, Sephirot and so on.
It is estimated that Euclid is the 2nd most published author in history; it seems geometry was more popular reading material in days one by then it is today! He is headed only by the Bible of course; another successful publishing story. Euclid’s Elements, little more than a book of dry old definitions, was taught in virtually every school in the Western world for over 2 millennia!
When there is a particularly pleasing form created from one of these definitions, a little artistic embellishment elevates it to a plateau of acceptance for 13-year-olds.
One such geometric theorem states that if you trisect the three angles of any triangle, an equilateral triangle is created in the center – a lovely accident? Divine artistry more like it! Draw three different kinds and color them beautifully as a feature page.
So finally one introduces a set of theorems on the Solids – the Ego aspect of this 4-fold exposition. These are prism, cubes, cones and especially their Egyptian derivatives, pyramids. These of course contain the triangle, as in a cryptic way does the cone – the following is a fascinating solids theorem:
A vertical section of a cone, when cut through the apex, always makes a triangle – 3 or more equal sections make a pyramid (triangular base, square, pentagonal – whatever!).
“Hey, I like that construction of the pyramid, I’m thinking of having one of those made for myself.” Interrupted the friendly Pharaoh just as Euclid almost had the last elusive element – the keystone indeed – of a new conceptual breakthrough on pyramids. “That’s really beautiful; by the way, what’s your name mean again?…nice? searching?…something like that – my Greek’s a bit rusty these days.”
“What?!”
“Er … I said what does your name mean?”
“It means quest for beauty!! You’re as irritating as fly up an ass’s nose – now let me get back to work!!”
“How dare you speak to your Lord that way! Your name certainly doesn’t match your nature! – To the croc pool with him!!… No wait, let him finish the theorem first.”
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