SECTION TWO

Chapter IV

THE MAGIC OF NUMBERS

We have stressed in an earlier chapter the fact that the curious properties of numbers are not to be considered merely a fertile field for mathematical recreations and jugglery.  The present chapter will further emphasize this by exploring some of the significant number relationships in occult studies.  We are now ready to present a more complete table of the numerical elements of the five regular polyhedra and relate them to significant number ratios in the revolutions of the planets of the Solar System and to the grand cycles in the history of the Root-Races on this earth.

First of all, let us inquire into the reason why there are five and no more than five of these interesting geometrical solids existing in nature.  Let us turn to Plate 4 and observe the order of these Polyhedra at the head of this Table: First, there is the Icosahedron, then the Octahedron, next the Tetrahedron, then the Hexahedron or Cube, and lastly the Dodecahedron.  Note in these figures a peculiarity of the numbers 3, 4 and 5.  Starting from the center and working toward the left: The Tetrahedron has 3 triangular faces about each of its vertices.  Next, the Octahedron has 4 triangular faces about each of its vertices.  Lastly, the Icosahedron has 5 triangular faces about each of its vertices.  If we were to try to construct a geometrical figure with 6 triangular faces about a vertex, we would find it impossible because it would result in a flat surface.  Now let us work from the center again, and this time to the right: The Tetrahedron has 3 triangular faces about each of its vertices; the Cube has 3 square (four-sided) faces about each of its vertices; and lastly, the Dodecahedron has 3 pentagonal (five-sided) faces about each of its vertices.  The number of lines bounding the faces have increased by 3, 4 and 5.  Again, if we were to try to construct a figure by placing 3 hexagonal faces about a vertex, we would once more have a flat surface.  This is the simplest method of demonstrating that there can be no more than five regular polyhedra.

It may be further noted from this table that just as the Icosahedron and the Dodecahedron are complementary, each having 30 edges, with the Icosahedron having 20 faces and 12 vertices, and the Dodecahedron having 12 faces and 20 vertices, so the Octahedron and the Cube are complementary.  Each has 12 edges; and whereas the Octahedron has 8 faces and 6 vertices, the Cube (Hexahedron) has 6 faces and 8 vertices.

At first there appears to be no complement to the Tetrahedron, because it has 4 faces and 4 vertices.  Actually, as demonstrated before, it is its own complement.  It interlaces with itself, forming the interlacing Tetrahedra around the Octahedron. 

Referring again to Plate 4, let us note that in line 4 is given the length of the edge of each polyhedron.  As the Cube represents fully developed Man in his relationship to the Universe, we are going to use the length of the edge of the Cube as the unit from which the edges of the other regular polyhedra will be derived.  Note the importance of the numbers of the Golden Section in relation to the edges of the Icosahedron and the Dodecahedron.  Let us take the lines in the order of their lengths; we shall find that they provide the first 4 terms of the numbers given in the infinite series of terms exhibiting the proportions of the Golden Section:

The length of the edge of the Tetrahedron is 1.414, or the square root of 2, since it is the diagonal on the face of the Cube.  The edge of the Octahedron is just one half the length of the edge of the Tetrahedron, which, therefore, equals .707.

We are now ready to embark upon one very important aspect of our study, which is to be found in the occult properties of numbers.  We should say at the outset that this bears little if any resemblance to numerology as a form of fortune telling or character analysis.  If we keep in mind the fact that this is a serious study, containing within it certain keys whereby we may become more cognizant of the laws of human and cosmic life, we can see at once that there is no room in it for superstition.  All aspects of this study must rest upon the solid foundation of esoteric teaching, or its value becomes nil.

It is necessary in some instances to run the risk of seeming to be arbitrary in the selection of certain numbers for specific purposes of our study.  A typical example of this is the statement to the effect that in occult reckoning 62 years is the ideal life span for a human being.  I cite this instance at the outset, because if I can make my position clear now, the rest of the study will be easier to follow.

A number of absurd inferences might be drawn from such a statement as the above, and we shall defeat our purposes if we let these stand in our way.  For instance, here are some things that we do not mean: We do not mean to imply that all people should die at the age of 72.  There are so many factors, which determine the length of life to be enjoyed by any one individual that we cannot readily set a limit on the desirable life of man.  Nor do we imply that a person who dies before the age of 72 has necessarily died before his time, or that a person living beyond 72 is living on borrowed time.

What exactly do we mean, then, and how do we arrive at the figure 72?  A suitable framework for our answer might be constructed in the following manner: The study of Mathematical symbolism reveals a very definite pattern.  The remarkable correlation between the geometrical forms, especially when interpreted in the manner that we have explained, enable us to extend our vision, as it were, into aspects of consciousness which are not ordinarily discernible.  The process by which this is accomplished is not easily explained.  However, it does come about through study and concentration.

To be more explicit: There are a large number of instances in which design in Nature follows the laws of mathematical harmony.  The spirals of some seashells are seen to be logarithmic spirals, or what are also called equiangular spirals, because at every point of the curve the angle between the curve itself and the line from the eye of the spiral is constant.  The mathematical properties of the spiral are well known, and it may well be taken as a pattern form the formation of the seashell.  There are many varieties of sea creatures having conical shells, and whereas the spirals may differ widely as among the various species, nevertheless each species exhibits its own pattern, with clearly defined characteristics.  Yet if you take any individual specimen and plot its own spiral, you are likely to find that it will approximate very closely the pattern upon which it is built, even though certain deviations will be observed.  This is only natural; it does not detract in any way from the significance of the observed pattern.

The mathematically minded student will find a fuller explanation in the Law of Probabilities.  An important aspect of this law exhibits itself in this manner: It describes the behavior of large quantities of things, but cannot predict the behavior of any given individual in the group.  For instance, it can predict with considerable accuracy the amount of noise reckoned in decibels that will be produced by a crowd of people at a football game, but it cannot predict that a certain individual will shout at any particular time. Even though there may be doubt about the amount of noise that any single individual is going to make, there is no doubt that there will be plenty of noise in the stadium.  That is the pattern of an afternoon at a football game.

In a later chapter, we shall consider some of the teachings about the Rounds and Races.  Here again we may be said to observe the law of probabilities in action, because we are dealing with very large groups of entities to which we refer as the hosts of monads which inhabit the Earth Chain; and whereas it would be impossible to take any one individual and state clearly just where he stands in the great process, yet, in considering the pattern followed by the group as a whole, we can state with certainty the laws governing the great processes of the Rounds and Races.

In natural design, the harmonious proportion of the Golden Section is of paramount importance.  Not only are its properties to be observed in some of the above mentioned spirals, but there are other remarkable manners in which this proportion occurs in nature.  There are many flowers in the daisy family, and they seem to exhibit a definite plan in the placement of the florets at the center of the flower, which later are placed by the ripened seeds.  If we take the largest member of the daisy family, the sunflower, because it is the easiest to observe, we see that the florets are so arranged that they form crisscrossing spirals; and the number of spirals turning to the left, divided by the number of spirals to the right, will give the number .618, already familiar to us as the proportion of the Golden Section.  This is so generally true that we are quite justified in stating that this is a pattern on which the flower are formed.  It would nevertheless be found upon close examination of any one particular flower that there are some imperfections in one or more of the spirals.  No great weight would be attached to this however, and we would not abandon the importance of the pattern on this account.

All five-petal flowers exhibit strongly the proportions of the Golden Section, for after all, are they not varieties of the five-pointed star, whose lines always cross at the points of the Divine Ratio?  As we observed in a previous chapter, it is just in the nature of things.  Yet see how much variety there may be found among a number of five-petal flowers!

Furthermore, in any five-pointed star, as in Fig. 7 in Chapter X, if we consider it to be constructed of five isosceles triangles but built upon a central inverted pentagon, we note that the vertex of each of these triangles contains 36 degrees, and each of the base angles contains 72 degrees.  This triangle is geometrically similar to the triangles forming the regular decagon that was generated by the moving circle (Fig. 4).  So it may rightly be said, regardless of individual minor differences, that the five-petal flower is built upon this all-important triangle, where in the base is to the side as .618 is to 1, while the vertex angles are 36 degrees and the base angles 72 degrees.  Thus we see the basic proportions which symbolize the building of a universe repeated in miniature in the formation of a simple wayside flower.

Now the angles as well as the lines in the Lesser Maze are exceedingly interesting also, as this complex figure very adequately exhibits a pattern.  We cited the case of the Cube, having as the sum of its plane angles 2,160 degrees.  This, we pointed out, is the same as the number of years required for the equinoctial point on the Earth's orbit (that is to say, the place seemingly occupied by the Sun at the exact moment of the Spring Equinox) to move backwards among the constellations of the Zodiac, through an angular distance of 30 degrees, or one-twelfth of the circle of the Precession of the Equinoxes.  We also pointed out that in the completed Lesser Maze the sum of all the plane angles is 15,120 degrees, or the same as the number of years required for the equinoctial point to slip backwards through an angular distance of 210 degrees, or 7 signs of the Zodiac.

Now this does not just happen; it reveals a pattern to us, and such an important one that we cannot dismiss it with the word “coincidence.”  Instead, we shall seize the opportunity before us and use the keys provided by a study of these regular polyhedra, and we shall come to some rather startling results.

It must be acknowledged that, just as in the cases of the design in flowers and sea shells, there is deviation in individual instances, which, however, does not negate the observed pattern, so we shall find that the numbers arrived at by modern astronomy for the Precession of the Equinoxes approach very close to, but do not exactly coincide with, the numbers suggested by Mathematical Symbolism.  Rather than cause concern, this should be something that we would expect.  After all, Nature is “regularly irregular,” and what a dreary world we would live in if all flowers were identical in size and shape, if all mountains had identical heights and slopes, and if we lived and breathed in strict uniformity. It is variety resulting in divergence from the underlying pattern which makes the universe interesting; and what is more important, it attests to a living universe.

Along this same line, it would be well to mention that the divisions of a circle into 360 degrees (credited to the Chaldeans) was probably no accident.  The remarkable relationships between the geometrical figures under study would not have been changed, but the numbers expressing these relationships would not have been so convenient, nor would the pattern have been so readily observed.

The number 360 ties in very neatly with the number of days in the year.  Here again we have an instance of an observable divergence from the pattern.  We are justified in seeing 360 days as the pattern or ideal number for the length of the year, even though we know that the year contains almost 365 1/4 days.  If we can accept this fact, and take it into account, recognizing the pattern for what it is, we will be in a position a little later to discover the pattern for the lengths of years of some of the other planets.  However, one step at a time.

Now, coming back to the number 72 as representing the number of years in the ideal or pattern for the life of man: We find an interesting correlation between man and nature, which will bear this out.  Furthermore, the average man in good health has a pulse beat of 72 per minute.  In the average human being in average health, and under average conditions, the rate of breathing is such that he draws 18 breaths per minute.  It follows from this that in 4 minutes or the time required for the Earth to turn 1 degree on its axis, a man has breathed 72 times.  On the grander scale, using the ideal numbers for the Precession of the Equinoxes, when a man has lived 72 years, the Processional Cycle has advanced 1 degree!

This number 72 will serve to introduce to us a large family of occult numbers, and in time we shall come to recognize them as friends as they appear time and again.  Some of them that we should learn to recognize right away are the all-important ones which are related to the cycles of the Zodiac, as for instance 2,160, as the number of years connected with 1 sign of the Zodiac, and 25,920, as the number associated with the complete Processional Cycle.

Another number with which we should become familiar at this time is 60.  This number is prominent in the regular polyhedra because there are 60 degrees in each angle of an equilateral triangle.  In the Lesser Maze there are no fewer than 36 equilateral triangles, giving a total of 108 angles, of 60 degrees each.

The Hexagon, composed of 6 equilateral triangles, has been shown to provide the outline within which all of the regular polyhedra may be drawn.  The hexagonal shape of the cells in the honeycomb, or again of the snowflakes, is so well known that we need not dwell upon this subject here.  Again, many flowers exhibit a six-fold pattern, in contrast to the fivefold as explained before.  One Sign of the Zodiac occupies 30 degrees; hence the angle subtended by two signs will be 60 degrees.  Since 2,160 years is associated with each sign of the Zodiac, twice that number, or 4,320, is of prime importance in the calculations of the cycles relating to the Rounds and Races, the Yugas, and so on.

Thus there are five important numbers in which we shall deal at this time.  These are 60, 72, 360, 2,160 and 25,920.

In approaching the matter of the planetary cycles in their relation to the geometrical figures, let us first consider how the important cycles are reckoned.  We learn by a study of the Brahmanical Tables relating to the evolution of the Solar System, the Earth, and the races of Humanity on our own Glove, that each cycle is preceded by a period of preparation, known as the Dawn; and following the cycle, there is a period known as the Twilight, during which the characteristics of the cycle just ended are receding, to be superseded by the characteristics of the new cycle.  The Dawn and the Twilight are each considered to be equal in length to 1/10^th of the cycle itself, so that we may begin to build up the numbers with which we are already familiar in the following manner:

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Plate V falls here
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The importance of the number 10, as observed in the moving circle as well as in the Tetraktys, gives us a very good starting point.  Furthermore, the prominence of the Golden Section in nature in general as well as in these studies specifically, prompts us to use this proportion as the working tool for building the cycles from the number 10.

Because the 5-pointed star exhibits so well the proportions of the Golden Section, we are going to use the number 5 to activate the process for cycle building with the number 10 as the starting number.  We shall include the Dawns and the Twilight's in our calculations:

The ramifications of the arrangements of these numbers are endless.  For instance, we can take any one of the important numbers in the column to the right and add to them the Dawns and Twilight's, and we have more of these interesting numbers, thus:

Many students will recognize this last number, 31,104, as the basis for the large number, 311,040,000,000,000, the number of years in the Life of Brahmâ, which represents the duration of the Solar System as given in the ancient Brahmanical Tables, dating back many centuries before Christ.

We are in a better position now to make a study of the last line of the Table of the elements of the regular Polyhedra.  This gives the sums of the plane angles of the various figures; and we shall now list them in the order of their magnitudes:

The first thing that may be noticed is that they are all multiples of 720, the number of degrees in the Tetrahedron.  They increase in the ratio of 1, 2, 3, 5 and 9, with a total of 20 x 720, or 14,400 degrees.

By this time we have become familiar with the number 72, and also with the important part played by the numbers 3, 4 and 5.  We may now observe that

3 + 4 + 5 = 12,   and   3 x 4 x 5 = 60.

Thus, 720 is the product of (3 + 4 + 5) and (3 x 4 x 5).

The numbers representing the sums of the angles of the regular polyhedra may be expressed in this manner:

There are certain characteristics shared by all of these occult numbers which make them easily recognizable.  One is that they are all divisible by 9.  This characteristic brings them all within a certain family group, we might say; and while a few of them stand out as being of special interest, it might be said, broadly speaking, that all numbers that are multiples of 9 are related to these occult numbers, however distant their connection might be.

The numbers that we built up using the pattern observed in the formation of the cycles as given in the Brahmanical Tables happen to be the most important ones, some of which are identical to those in the table of the angles of the regular polyhedra.  All of these numbers have many factors, the most important ones being 3, 4, 5, 6, 9, 10 and 12.  There are also larger factors, which are multiples of these, and they might be thought of as the building blocks out of which the occult numbers are constructed.

There is one number that is conspicuous because of its absence.  This is the number 7, which, curiously enough, has more prominence in our theosophical studies than any other.  From our earliest recollections of theosophical study, we have been taught about the seven Principles of Man, the seven Sacred Planets, the seven Globes of the Earth Chain, the seven Planes of Consciousness.  Why, then, does not the number 7 appear as an important factor in these occult numbers?  Well, the number 7 is there, but it is obscure, to be sure.  Most important, it appears in the sum of all the angles in the completed Lesser Maze (which most contain two Tetrahedra interlaced).  The number of degrees in this figure is 15,120, or the number of years related to 7 Signs of the Zodiac, whereas the Cube, which represents fully embodied Man, has 2,160 degrees, or the same as the number of years in the Messianic cycle, associated with 1 Sign of the Zodiac.

References to the Zodiac are necessarily sketchy at this time.  We give a fuller explanation of it in Appendix II.  It might be mentioned here that the term Messianic Cycle refers to the period of 2,160 years during which the spiritual influence of a Teacher associated with his own particular Sign of the Zodiac is felt, because such a Teacher guides the spiritual welfare of Humanity during that cycle.  At the commencement of another cycle, associated with another Sign of the Zodiac, there will be the appearance of a new Messiah, or Spiritual Teacher.  The appearance of these Teachers is timed to the cycles of the Zodiac.  As will be explained in a subsequent chapter when we are discussing the sixth and seventh Jewels of Wisdom, the teachings about the Hierarchy of Compassion and its work form some of the loftiest aspects of the Secret Doctrine.

One significant symbol connected with the nature of the spiritual Teacher may be appropriately mentioned.  Turning to Plate IV, we see that the Cube has been unfolded, and now takes the form of the Cross.  As the interlaced Tetrahedra enclosing the Octahedron were seen to generate the Cube, they are shown in their correct position in this figure.  Here we have an esoteric interpretation of the Crucifixion as a symbol of Initiation.  Rather than representing the physical death by crucifixion, it indicates that Man, as an embodied stream of consciousness, can and does at the appropriate times, undergo the mystical Crucifixion, a conquering of the gross “lower” self, and, making of himself a vehicle of a solar divinity, ultimately arises from the “tomb” of matter, a fully enlightened Teacher, a Savior of Mankind.  The particular type of Savior, known technically as an Avatâra, appears once in every Messianic Cycle of 2,160 years according to the Esoteric Tradition; and it is of the utmost significance that the Cross, no less than the Cube, contains within itself 2,160 degrees.

Tempting as it is at this time to embark upon a fuller explanation of the appearance of an Avatâra, this must be left to that portion of this book wherein we shall deal more fully with the nature of the Hierarchy of Compassion and the various types of great Teachers.

Probably the most important key to the system of Occult Numbers lies in the time periods of the planets.  We can tell from any book on astronomy that these time periods are a direct function of the distances of the planets from the Sun.  A student of the Ancient Wisdom, gradually becoming familiar with the occult pattern, will note that the majority of the planets seem to exhibit a rather loose adherence to this pattern.  If the Solar System were to adhere strictly to the pattern that seems to be in accord with what we have been studying, the distances of the planets would all have to be slightly different from what they actually are, and the orbits would have to be perfect circles instead of being elliptical as, indeed they are.  The odds in favor of the orbit of any planet being a perfect circle are so slight that it might be considered to be impossible.  By the same token, the likelihood of the placements of the orbits in respect to their distances from the Sun, so that their time periods would be exact multiples of one another, is again so remote that we may well consider that it would be impossible.  Nevertheless, the pattern is easily discernible; and we are going to imagine that the Solar System does present the ideal picture.  If this seems too arbitrarily conceived, we may take comfort in the thought that the ideal Solar System is not so very different from the Solar System as it actually is.  It all goes back to the Law of Probabilities, on which we touched earlier.  To the theosophical student, the divergence from the ideal pattern is not only to be expected; it shows that fundamentally the Universe is alive.  Such a statement is likely to horrify the scientist of the present day, but we need not apologize for the teachings on that account.

Seeing, then, that the lengths of the years of the planets approximate reasonably well with certain of the recognizable occult numbers, let us now make a table showing the length of years in terms of Earth days of certain of the planets known to Theosophical students as the Sacred Planets, and adjust their periods to the occult numbers related to them:

Some exceedingly interesting relationships become apparent from this table.  We find, for instance, that in such an ideal situation, Mercury would revolve around the Sun 4 times in every one of Earth's years.  Venus would revolve 20 times around the Sun during each of Jupiter's years.  Once in every 60-year-period the planets would all be in conjunction, or in syzygy, which means that they would all lie along a straight line with respect to the Sun.

In this, we have the real key to the Occult Numbers.  They are all built upon the relationships between the time periods of the planets.  As a starting point, take the years of Jupiter and Saturn.  Twelve Earth years make 1 year of Jupiter, and 30 Earth years make 1 year of Saturn.  12 x 30 = 360.  Five years of Jupiter equal 2 years of Saturn.  One year of Jupiter equals 20 years of Venus.

And so we could continue, finding these interesting relationships.  Actually, in this ideal Solar system, we would find that these planet time periods are all directly related to the sunspot cycle.  Specifically, there would be one such sunspot cycle for each year of Jupiter.

We shall now construct a table showing the numbers of revolutions of the planets during the important cyclical periods that we found previously, to wit: 60, 2, 360, 2,160 and 25,920 years respectively, and we shall see the continual recurrence of our important numbers as well as some others which are related to them.

In any one of these columns we may detect a number of very important ratios.  Of these we shall select the most meaningful and will show how the planetary time periods relate to the regular polyhedra, for they show the same numbers as the faces, lines and vertices.

The number 7 may indeed be found after its own manner in this ideal or pattern Solar system.  Just as Jupiter, taking its cycle from the Sun, seems to regulate the cycles of the other Sacred Planets, so it also seems to regulate in a different manner the cycles of Uranus and Neptune which, from the standpoint of the Ancient Wisdom, are in a somewhat different category from those known as the Sacred Planets, to wit: Mercury, Venus, Mars, Jupiter and Saturn.  Just now, let it be mentioned that 1 year of Uranus contains just 7 years of Jupiter, and 1 year of Neptune contains 14 years of Jupiter.

At our present level of learning and understanding, these ideas may be thought of as providing some of the letters in the occult alphabet.  As yet we have not developed the faculty of putting these letters together to form words and sentences which would give us that control over Man and Nature that typifies the genuine Adept; nevertheless, it may well be a step in the right direction.  Undoubtedly it is only after initiation that one learns how to make these teachings work for him in the sense of giving him the power and the right to control Nature's finer forces.