When we speak of the logarithm of any number, we mean the power to which we have to raise a certain fixed number called the base, in order to make it equal to the given number. For instance, we all know that 23 = 8. Here, 8 is the given number, 3 is its logarithm, and 2 is the "certain fixed number called the base." Thus 23 = 8 may be written: log2 8 = 3; or, expressed in words: the logarithm of 8 to the base of 2 equals 3. To make an algebraic statement, call x the given number; b, the base; and n, the logarithm of x. Thus bn = x. We may then state the following equality:
bn=x is the same as saying that log b x = n.
It is obvious that a complete system of logarithms can be constructed with any number as base, but the base 10 is used for the system or table of Common Logarithms. With ten as the base, we find, for example, that the logarithm of 8 is .9031, because 10.9031 = 8. Similarly, with base ten, the logarithm of 2 is .30103, because 10.30103 = 2. If 1.414 were to be used as the base, the logarithm of 2 would be 2, because 1.4142 equals 2. If 25 were to be used as the base, the logarithm of 2 would be .2153, because 25.2153 equals 2.
From the above it may be deduced that any number is theoretically the base of its own system of logarithms, which is merely to say that any number may be raised to the required power in order to make it equal to any other number; and, conversely, any number may be considered a power of any other number.
We learn in algebra that x° equals 1. This means that any number with a zero exponent becomes 1. Thus, taking the zero power of the three numbers used above as representing bases for three systems of logarithmic scales, 10° equals 1, 1.414° equals 1, and 25° equals 1. We find, then, that in each case, the logarithm of 1 is 0, or, to express it algebraically, regardless of the base, log 1 = 0.
With these facts in view, we may draw some interesting conclusions concerning Cosmic Life. The Ancient Wisdom affirms that the universe is constructed upon a hierarchical basis. All things are alive, and the universe is a living organism, composed of units, all of which in their turn are alive, and composed again of units. Thus is formed the Ladder of Life, of which every rung is a hierarchy. To put it otherwise, everything is composed of hosts of living units, subservient to, and uniting to form inner and outer vehicles of, the ruling consciousness; every entity is but one member of a family composed of hosts of like entities all subservient to, because uniting to form the vehicles of manifestation for, a greater entity. Thus, the idea of a supreme God, than which there can be none superior, fades away before the light of Theosophy, for it reveals such a being merely as the representative of a certain type of cosmic beings.
And now, "to moralize the subject," let us think of every entity in the universe as a spiritual force, a divine energy; and because every entity has a karman peculiar to itself, with no two karmans exactly alike, let us say that to every entity belongs a certain number, which number is the symbolic representation of the qualities and attributes which go to make that entity just what it is. Actually, numbers are merely names for relationships, and when we recognise that the universe is built upon the mysteries of number, we find that notation may be used to express symbolically the mysteries of the universe. Thus, it would not be surprising to find that in the Cosmic scheme the relationships and workings of the various intelligence-energies are, on their own plane, related to what we in our own limited way see to be numerical formulas and processes. Seeing the thing the other way around, then, we may expect to learn something of cosmic life by studying mathematics — symbolically.
Ascribing, then, to every entity a number, and keeping in mind that every entity is a hierarchy, let us call that number the base of a system of logarithms; the various progressive degrees of development of that entity are represented by the powers to which that base may be raised; while the resultant outer manifestation, on the physical plane let us say, is represented by the resultant number. Take man as an instance in point. Every man is the manifestation of a certain hierarchy, and the summit or Hyparxis is a consciousness-center, enduring throughout the cycles. That consciousness-center, then, is the entity to which we shall ascribe a number, calling it the base of its system of logarithms. The various stages of the cosmic journey will be represented by the various powers of that base-number, and the outer manifestation will be represented by the resultant value.
Our expression, "bn = x is the same as saying that log b x — n" becomes:
"A cosmic consciousness at a certain stage of the journey produces X, its outwardly manifested self, is the same as saying that the function by a cosmic consciousness of its outwardly manifested self, represents a certain stage of the journey."
This could be applied in numberless ways. Take mankind as a whole, "As many men on earth (as many values of x), so many Gods in Heaven (so many values of log x, representing the powers of the cosmic consciousnesses which are the bases or roots of the members of the human family)." The difficulty with us is that while we know something of the value of x, the personality, we know neither the base nor its power, our "logarithm." Thus, we cannot recognise either the Divine Monad or the Higher Self. The path of Self study leads to Initiation, the fruit of which is Self Knowledge.
Consider now the Solar System. The Cosmic consciousness, working through the sun and the planets, is the base of its system, the outer forms are x; and their higher counterparts are the logarithms of x, or the degree to which the consciousness center has progressed and succeeded in manifesting its innate powers.
We may extend our picture to universal proportions. We have observed that any number may be expressed as a certain power of any other number, and let us say of 10, for example. The universe is built upon a tenfold plan, and the entities therein might be considered as various powers of universal consciousness. Thus, the outer appearance of a thing is a maya, an illusion, and to discover its true nature we must find out the power of universal consciousness that that thing is.
Unity is in all things; from unity we issued forth, and when the great cycles of diversity are over, to unity shall we all return. This is symbolized in the fact that the zero power of all numbers is 1, or unity. As we have seen, it matters not what number we may take, its zero power is always equal to 1. Put it the other way, if you like: when the logarithm of x is 0, it matters not what the base number, x equals 1. Here we must conjure up the picture of the dawn of cosmic life. As yet there had been no manifested worlds, no entities to live on them; all was drawn into the "Great Mother," Space, the Heart of the Universe. All outer manifested forms had entered into union with the All. To put it in our mathematical form: throughout the Boundless, x equalled 1, because log x equalled 0. But the base remained unchanged. The essential core of every entity was a cosmic spark from the Central Fire, which had burned long ago in other cycles, and still retained its ultimate essence, its Divine Selfhood, even though it no longer manifested. But it was destined to pursue its way throughout new cosmic manvantaras, bringing forth from within itself new veils or garments, accumulating new richness of experience, until the close of the cycle, when log x will once more equal 0, and all outer forms be withdrawn into unity, universal Space, awaiting a new Cosmic Dawn.
Our conception of the Cosmic structure will be widened if we look upon every entity as three aspects of consciousness: firstly, the base of a hierarchical system, composed of entities below him on the ladder of life; secondly, the power, or indication of the degree of perfection of some entity higher than he on the ladder, which higher entity is the base of the hierarchy to which he belongs, and thirdly, x, the outer manifestation of an inner god, which is himself, belongs to him individually, as it were, and is his own future reward, if he will strive onwards and upwards for ever.
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