We are prepared now to step into the realm of “pure” mathematics, that is to say, mathematics for its own sake, and not seeking any practical or utilitarian application. The modern way of thinking is to view mathematics as a tool of science or industry; in fact numbers are not held to have any significance in themselves other than as a means to an end. The curious properties of numbers have long been recognized, but studies along this line are usually relegated to the realm of recreational mathematics.
I should like to stress the viewpoint with regard to Mathematical Symbolism that it is in no sense of the word recreational unless, indeed, it may be said to re-create in our hearts and minds an awareness of the profundity of the basic truths about Man and Nature. As we pursue this study, the various geometrical figures will be presented as a means whereby we may clarify our understanding of the teachings of the Ancient Wisdom. At this point, one might very well ask: What is Theosophy, or any other system of thought for that matter, to do with Geometry?
Let us put it in this manner: If the study of the geometrical figures to be presented here were confined to a strictly mathematical treatise, it would be a marvelous study in itself. The astonishing relationships between certain geometrical solids known as the regular polyhedra would excite wonder in the mind of any student; and it is to be regretted that so little attention has as yet been paid to these figures. Many students reading such a treatise would find that within their own minds, and without any suggestions from me, there would arise a feeling of awe, something akin to reverence. They might ask over and over again: Why should these things be? And lacking an answer, they would let it go at that, but might well have a vague feeling that there is more in this than appears on the surface.
Again, if such a purely mathematical treatment of the subject should fall into the hands of someone already versed in the teachings of Theosophy, he might well see for himself the relationships between these geometrical figures and the processes of world-building, the nature of Man, and his place in Nature. And he need not feel at all that his imagination is running away with him. Indulging in this kind of thinking will place him in company with some of the greatest minds known to the human race, Plato Pythagoras, and others.
Now it should not be thought that this is the only road to a deeper understanding of the teachings. This is an approach that appeals to a certain type of mind, a mathematically mystical mind, let us say. Other methods of approach will serve the purpose better to other minds and hearts. Pondering these differences can lead to an understanding of what H. P. Blavatsky must have meant when she said in The Voice of the Silence, “The path is one, the ways to reach the goal must vary with the pilgrim.” For, after all is said and done, the highest aim is not to find a complexity of meaning in the mathematical symbols; the aim and purpose is to find and ally oneself with the Higher Self. Our studies must point toward this goal; without this all-important objectivity they would have little value.
It will be seen from what has just been said that in a higher sense the study of Mathematical Symbolism does have a practical application; nor does this conflict with the opening words of this chapter, i.e., that we are now stepping into the realm of pure mathematics, with little utilitarian value in the ordinary sense of the word. If this study can help to bring to the student an awareness of the all-pervading Divine Life in which we live and move and have our being, then indeed it is an aid to his progress.
What, then, is a symbol? Is it something in which a truth is hidden so that it is not easily detected by those not entitled to it? In some instances truths have been concealed, or veiled at least, in myths and legends, and in parables, because there have always been the beginners in search of truth who cannot grasp as yet the whole meaning of the philosophy, and it must be given to them in such a manner as can be accepted. But we do a disservice to the reader if we entertain the idea that there are a chosen few who are privileged to receive the truth, while the majority, presumably because they are not yet worthy to receive, must be content with half-truths, or truths so cleverly concealed that they cannot be detected. This unfortunate manner of putting it has turned aside more than one potential student.
Therefore, let us phrase it in this manner: The truth is always available to anyone who is seriously in search of it. But at any particular time, it is available to the degree that the student himself has the faculty of taking it. It could be set forth in the fullest exposition conceivable, but if the student does not yet have the faculty of grasping it, he will not recognize it for what it is, and he will imagine that it is being hidden from him. Actually, he is hiding himself from it. When we do receive even limited teaching it is with the assurance that as our understanding grows, we shall receive it in ever-fuller measure. It can never be forced upon us, but of our own wills we can go forth and receive it.
Once this is clearly borne in mind, we discover that a true symbol is the most concise manner in which a truth can be revealed. Not all truths are conveyed in words. They do not need to be. The truths existed long before speech was developed. In fact, the ultimate way in which they may be transmitted is through a type of communication that transcends words and requires in the pupil the faculty of intuitive understanding. This is what every student should seek to develop.
Now we shall come in time to regard the Dodecahedron as a symbol of manifested Nature, or Prakriti. This is because in its relationships to the other regular polyhedra, it demonstrates the most concise manner in which the truths about the manifested universe can be presented. If we press the question: Why is this so? We can only say, “It is in the nature of things.” There is no other reply; and this applies to all the symbols that will be presented. Why are we conscious? It is in the nature of things. Why is the Boundless the source of all that is? It is in the nature of things. Why are we embarked upon an age-long pilgrimage back to the source from which we sprang?
Again we must say, it is in the nature of things. Why does a moving circle generate a series of marvelously related geometric forms? Once more, it is in the nature of things. Why do these forms represent the journey of the Eternal Pilgrim back to the Home from which he issued forth? It is in the nature of things.
Having thus set the stage, I would like to quote from the opening passage of
H. P. Blavatsky's Proem to her Secret Doctrine.
PAGES FROM A PRE-HISTORIC PERIOD
“An Archaic Manuscript — a collection of palm leaves made impermeable to water, fire, and air, by some specific unknown process — is before the writer's eye. On the first page is an immaculate white disk within a dull black ground. On the following page, the same disk, but with a central point. The first, the student knows to represent Kosmos in Eternity, before the re-awakening of still slumbering Energy, the emanation of the Word in later systems. The point in the hitherto immaculate Disk, Space and Eternity in Pralaya, denotes the dawn of differentiation. It is the Point in the Mundane Egg, the germ within the latter which will become the Universe, the ALL, the boundless, periodical Kosmos, this germ being latent and active, periodically and by turns. The one circle is divine Unity, from which all proceeds, whither all returns. Its circumference — a forcibly limited symbol, in view of the limitation of the human mind — indicates the abstract, ever incognisable PRESENCE, and its plane, the Universal Soul, although the two are one. Only the face of the Disk being white and the ground all around black, shows clearly that its plane is the only knowledge, dim and hazy though it still is, that is attainable by man. It is on this plane that the Manvantaric manifestations begin; for it is in this SOUL that slumbers, during the Pralaya, the Divine Thought, wherein lies concealed the plan of every future Cosmogony the Theogony.”
Again, quoting from page 4:1
“The first illustration being a plain disk, the second one in the Archaic symbol shows a disk with a point in it — the first differentiation in the periodical manifestations of the ever-eternal nature, sexless and infinite ‘Aditi in THAT’ (Rig-Veda), the point in the disk, or potential Space within abstract Space. In its third stage the point is transformed into a diameter [Fig. 2]. It now symbolizes a divine immaculate Mother Nature within the all-embracing absolute Infinitude. When the diameter line is crossed by a vertical one, it becomes the mundane cross [Fig. 3].”
Referring to the explanation of the Three Logoi in Chapter One, we may indicate that the plain disk represents the Boundless, and as it lies upon a dark background on the original parchment as described by H. P. Blavatsky, duality is suggested, and represented in our studies by Parabrahman (outside the disk) — Mûlaprakriti (within the disk). The point within the circle now represents the First Logos, Brahman-Pradhâna, the point indicating differentiation, although as yet no manifestation is apparent.
When the diameter appears, we now have a representation of the Second Logos, Brahmâ-Prakriti or Svabhavat, semi- or quasi-manifestation (Fig. 2). The Third Logos, Mahat, is represented when the vertical line crosses the diameter. But note that this is not a static symbol. In Fig 3, the vertical line ends in an arrow, indicating motion. The circle will be made to descend, representing the action of the Third Logos in manifestation. In the occult representation of the Cross, the horizontal diameter represents the feminine, and the vertical line the masculine aspects of Svabhavat.
On page 29 of The Secret Doctrine, note verse 10 of Stanza III of the Book of Dzyan:
“Father-Mother spin a web whose upper end is fastened to Spirit — the light of the one darkness — and the lower one to its shadowy end, matter; and this web is the universe spun out of the two substances made in one, which is Svabhavat.”
Let us now observe that the horizontal line has been lettered AB, and since it is taken to represent the feminine energy, let us see what it will accomplish. Turning to Fig 4, we see that the circle has descended a distance of its own diameter. The stage is now set for the springing into life of the seeds of manifestation brought over from previous cycles. This is represented by the new position of the circle. It no longer intersects any portion of its old position; the latter now represented by the broken-line circle. In other words, the circle occupies an entirely new position in space.
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Fig. 2 — The Diameter Appears.
Fig. 3 — The Mundane Cross.
Fig. 4 — "Mother-Father Spin a Web".
Point A on the original horizontal line is considered to be fixed in space, and the diameter has thus become a vector; and because it must still pass through the center of the circle, it takes its new position as shown in Fig. 4, and is lettered, as before, AB. The phenomenon we are about to perceive would have been the same if we had taken Point B as the fixed point. The result then would merely have been a reflection of what has occurred.
The lengthened line AB is cut by the circle at the point K, and a remarkable relationship is now seen between the line segments. This may be expressed thus: The shorter segment, AK, is to the longer, KB, as the segment KB is to AB. This curious property is known as the Extreme and Mean Ratio, called by the Greek Scholars the Golden Section. We may state this numerically thus: The radius of the circle is considered to be unity, or 1. Thus, the diameter will be 2. The length of the line segment AO (to the center of the circle) is equal to the square root of 5, or 2.236.2
Cutting AO at the point K has the effect of reducing AO by the length of the radius of the circle, so that AK will equal 2.236 — 1, or 1.236. So now we have: AK equals 1.236. KB (the diameter) equals 2, and the line AB will therefore equal 3.236.
The proportion may then be stated thus:
1.236 : 2 = 2 : 3.236, which is the numerical way of saying that the smaller line-segment is to the larger as the larger is to the whole line.
The numbers will reduce to a more convenient form by dividing by 2, and we then have:
.618 : 1 = 1 : 1.618
These terms, .618, 1, 1.618 are only three in a series that extends endlessly in either direction, each term bearing exactly the same relationship with the one following. A few of these terms that will be familiar to students of geometry are as follows:
.382, .618, 1.00, 1.618, 2.618, and so on.
Additional material on the Golden Section, as noted, will be found in the Appendix, but it will suffice here to say that much research on the part of many scholars has been done with this remarkable property of the Extreme and Mean Ratio, and it has bee found that there are many instances of design in Nature that follow this interesting proportion. So revered was it by the ancient Greeks that they used it widely in their architecture and sculpture, and this is one of the reasons that the beauty of Greek art is so well known and loved. This proportion is also used by some modern architects. We are about to discover another kind of beauty in it.
Having divided the numbers by 2, we can accomplish the same thing geometrically by dividing the line segment AK at the midpoint and taking the ratio of MK with respect to the radius of the circle. Thus we have: MK: KO equals .618:1. This length, MK, is seen to step exactly ten times around the circumference of the circle, and we have inscribed a regular Decagon!
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Plates III and IV fall here
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This is a wonderful fact of geometry. First of all, it suggests that 10 is a natural number to be used as a base for notation. In modern concepts of notation, other bases are being used, and they have their value for the purposes for which they are intended, however, these present studies will probably convince the student that there is a naturalness about the number 10.
Now, it may be a bit confusing at times to find Theosophical students using a wide variety of numbers in explaining the doctrines. We sometimes come across expressions something like this: “The planes of consciousness may be numbered as 7, 10, or 12, according to the manner in which the teaching is being presented.” I have sometimes wondered if our good listeners and readers are waiting for us to make up our minds! Well, the reason for this seeming confusion is not really so difficult to find. It all depends upon how much of the teaching is being explained at any one moment. Let us be specific in one instance.
It is sometimes convenient to allude to the threefold division of man's nature as body, soul and spirit. This threefold division is usually attributed to St. Paul. As far as this goes, it is perfectly correct and is full of profound meaning. But if we wish to elaborate, we turn to the sevenfold division of man's constitution, as explained in many of our textbooks, presenting in this manner the seven principles of Man. Then, when we wish to go still deeper into the mysteries of man's consciousness, we explain that these are the seven manifested principles, but that above these (speaking diagrammatically) there are three unmanifest principles, about which we seldom speak for the reason that it is next to impossible to describe them, however important they may be. In the first place we stumble over the inadequacy of words when we speak of unmanifest principles, or again of unmanifest globes. If they are unmanifest, how can they have the shape of globes? Well, they don't really, but we do not have the words for them and we do the best we can with diagrams.
Then, again, when we speak of 12 as the complete number, we mean the hierarchy of 10, with the upper link, which binds it to the hierarchies above, and the lower link, which binds it to hierarchies below. These last two, completing the full number 12 are not really principles at all, but they are very important for our study when we take the enlarged picture of the hierarchical structure of Nature and see that there are endless planes of consciousness above and below ours. We must think of Man himself as a stream of consciousness which takes its rise in planes of existence far higher than any in which he has conscious awareness; his stream of consciousness extends beyond his own life and touches other realms. But that is another story.
Thus the circle in motion, generating a regular decagon, may be taken as a most graphic symbol of the formation of the ten planes of consciousness within the Universe.
The principles underlying these numbers appear in other forms. Perhaps of paramount importance is that of the Tetraktys. This was the symbol probably most revered by the Pythagoreans. An oath taken on the Holy Tetraktys was held to be binding for life. This single triangle of ten dots stood for them as a symbol of creation; and fore good reason, as we shall see. In its usual form it consists of ten dots arranged in the form of an equilateral triangle, thus:
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It may appear sometimes in other forms.3
In Fig. 5, we discover that the triangular representation of the Tetraktys is an exceedingly important one and exhibits a very curious property. It shows quite graphically what we mean when we speak of the seven manifested planes and the three unmanifested. The three dots which are the vertices of the triangle stand apart, and only the seven dots at the center of the figure will be used in a large number of the diagrams which are to follow.
The manner in which these dots are employed in this figure has a story connected with it. It relates to the Old Norse legend concerning Thor's hammer. According to the story, Thor wanted to have a hammer that could be used for wielding the thunderbolt, and he went to the smith Hafnir, who agreed to make the hammer. While Hafnir was working, the dwarf Loki, who was a troublemaker, changed himself into a gadfly and stung Hafnir on the forehead until the blood ran down into his eyes and blinded him. Hafnir raised a hand from his work to brush away the blood, and in that instant the work was stopped and Thor's Hammer was not completed, nor does the legend tell us what the hammer would have been like. But the story does hold adumbration's of a deep mystery, for in showing the Hammer in the familiar form of the Svastika, as it is customarily depicted, we do have a symbol of cosmic rotation; and it is significant that this symbol has been used by many cultures. May we not suppose that the Tetraktys furnishes the key to the mystery?
The conventional Svastika, as everyone knows, is made in the form of the Cross-with the arms bent at right angles — a two-dimensional figure. The completed Thor's Hammer, however, becomes an even more meaningful symbol. This we show in Fig. 5, where the diagram, though two-dimensional, may be used to represent a three-dimensional object. Note that six of the dots stand equidistant from the central seventh. By joining the three pairs of dots through this central seventh we have three lines, to be considered mutually at right angles to each other, and representing the three axes of rotation by virtue of their being bent.
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All objects, whether they may be atoms or stars, are subject to three axes of rotation. In the case of a ship, or a space vehicle, these three motions are called roll, pitch and yaw. The three motions of the Earth are: its rotation about the axis running through the north and soul poles; the motion at right angles to this, which determines the inclination of the axis; and, finally, the third motion, at right angles to the second, which determines the “precession.” In Appendix II we shall give a detailed account of the Precession of the Equinoxes, a teaching — or fact of astronomy, rather — with which every student of Theosophy should be familiar. If it were not for the three axes of rotation, there would be no Precession of the Equinoxes, with the wonderful correlations between the cycles and the geometrical figures shortly to be studied and the deep philosophical implications inherent in the numbers related to them.
We may introduce, at this point, one more facet of our study of the Tetraktys: If we were to take five of these three-dimensional Svastikas and place them concentrically, observing, of course, the correct angles between the radiating lines, we would find that the bent arms would of themselves form a most interesting geometrical figure, the Icosahedron. See Plates 1 and 2. This is a regular Polyhedron, having 20 triangular faces and 12 vertices. We have now only to change the positioning of the five Svastikas, and the bent arms are seen to form a second geometrical figure, the Dodecahedron. This is a regular polyhedron consisting of 12 faces, all of them being regular pentagons and having 20 vertices. A detailed study of these and other figures will be taken up in the succeeding chapters; they form the main body of our study. The highly important point being brought out at this time is that this mystical Tetraktys is actually the link between the laws of mechanics and the study of Mathematical Symbolism. It serves as a magical bridge joining mystical thought with the more pragmatical concepts of natural laws.
In concluding this chapter, we should emphasize that while each and every mathematical fact that will be brought forth can be demonstrated by standard methods of geometrical proof, such proof nevertheless merely places the stamp of approval, as it were, on the mechanics of the study. It does not really explain why these things should be. An effort to understand the “why” carries us much deeper than the laws of mathematics as such, for it forces us to develop faculties which transcend the intellect. Although we may reach to a certain kind of intuitive understanding, we cannot always formulate such comprehension in words; and once more, when we attempt to answer the great question WHY, we must be content with the simple reply: “It is in the nature of things.”
1 See Fig. 1, Fig. 2 and Fig. 3. In The Secret Doctrine, these figures are interspersed within the text but for the convenience of the reader, these are set forth here as separate figures. I have added the downwaard-pointing arrow to the cross in Fig. 3.
2 For a full explanation of this, Appendix I.
3For instance, those familiar with the Brahmanical Tables giving the number of years in the four Yugas or great ages will recognize in mumerical form the same relation: 1 - 2 - 3 - 4. Thus, the shortest is the Kali-Yuga, with 432,000 years: the next is the Dwâpara-Yuga, twice as long, with 864,000 years; then comes the Tretâ-Yuga, this having three times as many years as the Kali-Yuga or 1,296,000 years; and lastly is the Krita-Yuga, or Golden Age, four times as long, with 1,728,000 years. Added together, these make up the Mahâ-Yuga of 4,320,000 years, just ten times as long as the Kali-Yuga. This statement is actually an over-simplification; it would apply to the Third, Fourth, and Fifth Root-Races more than to the First and second, which are much longer, or again, to the Sixth and Seventh, which are shorter.