The purpose of this book is to develop and apply an arithmetic of qualities. The study of the qualitative aspects of numbers is called Mathesis. This is an old word used here in a new meaning. The word quality is derived from the Latin word qualitas which means the "what kindness" of an object. This purposed will be achieved by deducing the basic qualitative relationships among the digits. These relationships will then be shown to be similar to other qualitative relationships. This similarity of qualitative structure forms the basis of practical mathetical applications.

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Numbers and Qualities
  Figurate Numbers
  Triangular Numbers
  Square Numbers
  The Tetraktys
  Decimal Octaves
  The Digits
The Cabala
Modern Numerology

Numbers and Qualities

In this world there are a vast number of things. The things themselves are inescapable and obvious. But there are so many things that we can not possibly hope to understand and deal with each one individually. Fortunately, we possess minds that fit these things into classifications and categories. To do this, relationships among things must be discovered. Relationships are the most important ideas we can have about things. This is because the same relationships may apply to several different sets of things. This means that past experience can be used to anticipate the future. If the relationships we postulate or surmise are so accurate that we can anticipate the future in detail, then we have a science.

Mathematics is universally regarded as the queen of the sciences because mathematics is the study of symbols and their uses. Symbols are things which can carry a wealth of meaning. And the mathematician concentrates on the relational meaning of the symbols. Mathematics is primarily a tool for thinking clearly about numerical or logical relationships among things. Number enables us to calculate an unknown quantity by using already known values. Logic is the means for determining the relationships among ideas and for finding the more important or basic ideas. It is necessary to realize that mathematics does not discover or verify fact about the external world. Mathematics, and this includes mathesis, deals with an inner world of symbols and relationships. By providing easily used symbols, mathematics enables us to think clearly about quantities. Anything that can be measured or counted can be analyzed to discover new relationships.

But numbers seem to have qualitative meanings in addition to their purely quantitative significances. As Carl Gustav Jung, the great Swiss psychologist, once wrote:

Number helps more than anything else to bring order into the chaos of appearances. It is the predestined instrument for creating order, or for apprehending and already existing, but still unknown, regular arrangement or "orderedness" . . . It is generally believed that numbers were invented or thought out by man and are therefore nothing but concepts of quantities, containing nothing that was not previously put into them by the human intellect. But it is equally possible that numbers were found or discovered. In that case they are not only concepts but something more - autonomous entities which somehow contain more than just quantities."

Numbers certainly denote quantities. But they also seem to convey qualities. The quantitative uses of numbers have been explained extensively. But still the qualitative uses of numbers have not been developed.

A qualitative order in numbers will provide a much-needed tool for clear thinking about qualities. The basic qualitative question is "What kind of a thing is this?" This question implies a system of classification and arrangement. A classification must be logical and consistent within itself. It must fit the facts as nearly as possible. Each field of thought has grown its own classification scheme. Books in libraries are filed one way; stars are cataloged by different methods. The language we speak has grown an inherent classification system which we never even think about. A general mathematical and symbolic system is needed for handling qualitative classifications such as these. Showing that such a system exists and showing how it can be used are the purposes of this book.

The ideas and results in this report have been discussed in as simple and straightforward manner as possible. The reader of a book on numbers can be trusted the follow the details which involve a little arithmetic. A note of caution must be added concerning some of the terms used to convey qualitative ideas. The names of the alchemical elements - Air, Fire, Water, and Earth - are used extensively, but not in their ordinary meanings. These terms are used to imply a fourfold division of the Whole and to imply certain relationships among these parts. Chaos is used to mean something which we can not classify or analyze. It necessarily must imply that chaos to one person may be a cosmos to another person. Cosmos is the opposite of chaos. A cosmos is something that is orderly and well organized.[top]


Pythagoras was evidently the first to envision a universal numerology, a final and complete statement of the cosmos in terms of numbers. He was born on the Isle of Samos in the Grecian Archipelago in 584 BC and died in 495 BC. He traveled widely in Egypt and Babylonia and even perhaps went into India. He must have learned much from the Babylonians. They were the master mathematicians of antiquity. These searchers of the skies also had long had symbolic interpretations of numbers. For example, they assigned numerical equivalents to the gods for purposes of divination. Returning home, Pythagoras established a secret school and society in Crotona, a Greek colony in southern Italy. The political activities of his society led to a mob's burning the school and Pythagoras with it.

Pythagoras' contribution to science is the concept of explanation by mathematics. Before his time mathematics was only a tool for measurement, surveying, and inventory keeping. After his time the philosophers and scientists have sought to develop explanations for natural occurrences in mathematical terms. This is a vastly different concept from that of merely weighing or measuring. Mathematics has emerged as a sort of super-science. And Pythagoras was the first of these scientists. To be sure, he made his mistakes. But he was not mistaken in his first principle: "All things are (explained by) numbers."[top]


Legend has it that one day, while walking past a blacksmith's shop, Pythagoras noticed that some hammers rang with melodious concords, some range with discords. Borrowing the hammers, he took them home and performed the first recorded experiment of science. He hung the hammers with strings from a lyre and found that those hammers with simple ratios between their weights produced pleasing concords. Later he formulated the numerical ratios for the octave which are presented in Chapter Two. {It must be noted that the legend is nonsense - the sound produced by a hammer's striking does not have a simple relationship to sound produced by hanging that hammer from a string and then plucking the string!}

Having shown that musical harmonies are determined by numbers and are not just a matter of personal good taste, Pythagoras began to study numbers in the hopes of finding more such laws. He and his followers made many of the discoveries in geometry which form the subject as taught in high schools. Pythagoras also taught that the planets revolved around the central sun with varying speeds. His genius was fettered by being so far ahead of his age. There were no instruments for making accurate measurements of the motions of the planets, no instruments for measuring time. Worse, the system for writing and manipulating numbers was extremely difficult to learn and use. Perhaps his astronomy would have been as perfect as his harmonics had he possessed the tools of Brahe and Kepler.[top]

Figurate Numbers

Pythagoras discovered the vital system of polygonal numbers. These numbers form a link between geometry and arithmetic by showing that a certain number of dots can be arranged into a geometric figure. Because of this link, some numbers are triangular, square, pentagonal, and so on.[top]

Triangular Numbers

The first few triangular numbers are as follows.

     1      3       6       10          15
     o      o       o        o           o
           o o     o o      o o         o o
                  o o o    o o o       o o o
                          o o o o     o o o o
                                     o o o o o

To continue the sequence, the general rule is that the numbers in succession are added to form the sequence.

  1 = 1
  3 = 1 + 2
  6 = 1 + 2 + 3
 10 = 1 + 2 + 3 + 4
 15 = 1 + 2 + 3 + 4 + 5

Square Numbers

The square numbers are more familiar. The first few squares follow.

     1      4       9       16          25
     o     o o    o o o   o o o o    o o o o o
           o o    o o o   o o o o    o o o o o
                  o o o   o o o o    o o o o o
                          o o o o    o o o o o
                                     o o o o o

This sequence is formed by adding only the odd numbers in succession.

  1 = 1
  4 = 1 + 3
  9 = 1 + 3 + 5
 16 = 1 + 3 + 5 + 7
 25 = 1 + 3 + 5 + 7 + 9

There are many interesting theorems about these figurate numbers. For example, the sum of two successive triangular numbers is always a square number. Figurate numbers are useful tools in the study of crystal forms and molecular structures. It can be shown that there are just 32 possible crystal forms and 230 different ways of hooking together arbitrary shapes in space.[top]

The Tetraktys

                            o o
                           o o o
                          o o o o

The number ten arranged in triangular form as above is the Holy Tetraktys of the Pythagoreans. To them it was the supreme symbol of God, and to it they addressed their prayers.

Bless us, divine number, thou who generatest gods and men! O holy, holy Tetraktys, thou that containest the root and source of eternally flowing creation. For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-abounding, the first-born, the never-swerving, the never-tiring holy Ten, the keyholder of All.

The importance of the Tetraktys lies in two things. First, because the Greeks named dimensions differently, they saw in it all the possible figures of geometry and the universe. They said

One point defines a point;
Two points not on the same point define a line;
Three points not on the same line define a plane;
Four points not on the same plane define a space.

Now we diminish each of these numbers by one and say that space is three-dimensional.

The second significant aspect of the Tetraktys is that is shows the whole (=10) divided into four parts. Perhaps it is from this that the doctrine of four elements composing the universe was derived. Empedocles, who originated the four-element theory, was certainly familiar with the Pythagorean ideas since he lived only a few miles from the school at Crotona.[top]

Decimal Octaves

Another very important doctrine of the Pythagoreans was the law of Decimal Octaves. As the Pythagoreans said, "The decade (=10) contains all things, since numbers beyond the decade merely repeat the first ten." The doctrine thus states that a large number is a higher octave of the digit occupying the unit's position. For example, 25 is basically a 5 modified by the 2 in the ten's position. The most specific number is the digit in the unit's position because this definitely indicates the digit within the decimal octave. The other digits in the ten's, hundred's, and higher places influence the overall significance of the total number.

The zeroth decimal octave (that is, the single digits) contains the most and important and basic numbers just as zero itself is the most important and basic digit. For example, 25 is the 5 of the second decimal octave. The zeroth octave is the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The second decimal octave numbers all have a 2 in front: 20, 21, 22, etc. The tenth octave would be in hundreds: 100, 101, 102, etc. This is a very logical way of doing things. One can only regret that the same practice was not followed in naming the centuries by starting with a zeroth year in the zeroth century.

This idea of decimal octaves later degenerated to the practice of "theosophical reduction," a cabalistic idea which has nothing to do with the Theosophical Society of India. To apply theosophical reduction the digits are added together, repeatedly if necessary, until only one digit remains. Taking 25, the cabalist would reduce this by adding 2 and 5 to make 7. He would then affirm that 25 equaled this 7. This method of reduction can completely destroy any significance of this original number. For example, applying theosophical reduction to 17 produces 8. This makes an even number (=8) from an odd number (=17). But to the Pythagoreans, the distinction of odd and even was the most basic attribute of number.[top]

The Digits

Since larger numbers are combinations, and higher octaves of, the first ten numbers, the Pythagoreans developed symbolic and qualitative meanings for the ten digits as follows.

The Pythagoreans did not know a zero, but we can identify it with the Chaos, the origin of categories and the universe.
This is the qualitative equivalent of ten.
signified essence, self-hood, individuality.
otherness, difference, diversity, indefiniteness
mediation, atonement, completeness. In Greek mythology the rule of the world was divided among three gods: Zeus ruled the air, Poseidon the sea, and Pluto ruled Hades. A trinity identified Each god: the three-forked lightning, the trident, and the three-headed dog Cerberus.
indicated squareness, justice, fair dealing
the first combination of the odd (=3) and the even (=2) symbolized marriage; also life, health, and vitality.
the number of luck or chance
the number of the entire cosmos, three being the deity and four indicating the soul, 7 = 3 + 4.
stood for solidarity and firmness since it is the only cubic digit, 8 = 2 x 2 x 2.
represented the limit of numbers since 10 was merely a repetition of 0

And, in general, the even numbers were considered more passive, feminine, changeable, and earthly. Odd numbers such as 1, 3, 5, and so on, were masculine, active, firm, and of a celestial nature.

This particular idea was certainly not held only by the Pythagoreans. The Chinese had identical beliefs. The odd numbers were considered manifestations of Yang, the masculine or celestial influence. The even numbers showed Yin, the feminine and earthly influence. Incidentally, Marcel Granet's book listed in the Bibliography is a prime source for information about Chinese numerology. This subject has not been treated here because the oriental numerology never tried to explain or analyze the cosmos. Oriental numerology remains on a par with recreational mathematics in the West. Interesting coincidences are pointed out, but no general synthesis is attempted.[top]

The Cabala

The teaching of the Pythagoreans became the common property of all of who wished to know them. So it was that when the Roman Empire gathered up all the Mediterranean peoples they could use or abuse the Pythagoreans theories as they saw fit. The Greeks and the Hebrews both used every letter in their alphabets as numbers. Every word was a number and any number could be made into a word. The Hebrews already had various methods of interchanging letters in the text of scriptures so that any passage could be altered to say anything that it did not say originally. To this native tradition of altering texts the scholarly Hebrews added the newer ideas of the Pythagoreans and came up with theosophical addition and reduction.

This concoction of miscellaneous numerical doctrines and the beliefs of Christianity caused a good many strange tenets of the Gnostic religion. By the time the orthodox church suppressed the Gnostics, the teachings of Pythagoras had become almost completely adulterated. It is only recently that modern scholarship has separated Pythagorean doctrines from their gnostic admixture. But it was precisely this admixture that was transmitted to the Middle Ages. As E. T. Bell rightly says, the numerology of the Middle Ages was a smashing comedown from that of the Greeks.

The most important numerological idea from those times was the cabala. The cabala uses the gnostic or Zoroastrian doctrine of the emanation of rays from God. In fact, the cabala is mainly a mystical, not a numerological, field. But, in so far as it claims to be a study of the qualitative aspects of numbers, we must examine it in more detail. As a work of mysticism it may be sublime, but it has as least three serious logical defects beside theosophical reduction.

The first of these defects is that there is no analysis of the digits at all. These are assigned in straight numerical order to both the letters of the Hebrew alphabet and to the ten sephiroth or qualities. We must analyze numbers and the sounds of speech, and then - if a sequence is wanted - it can be arranged by the relative weight or the elements of the digits.

The second defect is that some numbers are said to be active, some more passive, and a neutral set reconciles the actives and the passives. But actually, all pairs of complementary numbers are reconciled or harmonized by the same neutral number just as all pairs of complementary colors combine to form the same neutral color.

The third defect, but not less important than the first two, is that the cabala can provide a septenary (seven basis) system of categories only with great difficulty. To get a set of seven from the ten numbers, the cabalist puts 1, 2 and 3 into one palace or category, 9 and 10 into another palace, and lets 4, 5, 6, 7, and 8 each form a separate palace. Chapter Two of this report demonstrates the necessity of having a septenary system with subsets of three and four members within a general decimal scheme.[top]

Modern Numerology

At present the old Pythagorean number mysticism and the cabala have married and produced an offspring known as numerology. Or rather, they have produced twin children both known as numerology. The basic use of numerology is character reading or numerical psychoanalysis. Why its customers could not sit in front of a blank television set and come to a better understanding of themselves is hard to discover.

The way in which this numerical psychoanalysis is done is exceedingly simply. A digit is equated with each letter of the alphabet, so any name can be written as a group of digits. These digits are then added together, and theosophical reduction is applied to the sum until finally a single digit is found. This digit is the numerical equivalent of the whole name. The qualitative description of that digit fits the name and thence the person who bears the name. Sometimes the vowels and consonants are added separately to give two more possible digits from a name. Then also the date of birth is written in numerical notation, and these numbers are added and reduced to a single digit.

But numerologists can not agreed on which numbers to assign to each letter of the alphabet. Those who like things to be simple just give the letters the numbers 1 through 9 in straight numerical order. A = 1, B = 2, ..., Z =26 = 8. No attempt is made to spell the name phonetically, and all the letters are counted whether they are silent or not. Those who like to complicate matters given the letters of the Hebrew alphabet numerical equivalents as before. To apply their scheme to English, they have to translate the Hebrew letters into the English pronunciation. This is almost impossible for two reasons. First, the Hebrew alphabet has no vowels. Second, the consonants are not all like those of English.

So it is that there are two camps of numerologists: the Roman alphabet cohort and the Hebrew alphabet tribe. However, they are both agreed upon the use of theosophical reduction to produce an ultimate result of a single digit for their labors. In both camps many different methods of calculating characters and future events from names and dates have been developed. The reader is referred to the books by Sepharial, Boylile, Montrose, Campbell, and de St. Germain listed in the Bibliography. The two books by E. T. Bell provide a witty historical introduction to the whole subject of numerology and numerical qualities.[top]