by Elizabeth Butt

1.

RECENTLY the British began the complicated process of adopting the metric system (I strongly believe this country should do likewise.), and the change-over may be a long and troublesome one. The first step was transferring from the use of Fahrenheit to Centigrade scales for temperatures. For some years, the older folk may feel considerably colder at 0 degrees (C) than they would at 32 degrees (F); but, of course, it is all the same. It will take some getting used to.

Numbers have their own fascination, and it is interesting to ponder on how differently civilizations might have developed had no one discovered, or invented, the concept of *zero*. That may have been a greater boon than the discovery of the wheel – or at least, equal to it.

Around 650 B.C., there was a great advance in arithmetic and geometry, partly due to the introduction of Egyptian papyrus into Greece. Before this time, geometry had been taught and studied by drawing diagrams in sand scattered on the marble floors. The invention of printing in the 15^{th} century was hardly a more important stimulation to thought than the introduction of writing material.

Pythagoras is usually ranked in history as the first of the great mathematicians of ancient times. Little is known of his personal history. He was born (as they then dated) between the 50^{th} and 52^{nd} Olympiads, or between 580 and 568 B.C. Although he was called a Samian, because he did live on the Island of Samos as a young man, it is not known that he was born there.

Samos, at that time, was becoming a center of Greek art and culture. It was a rich era of development throughout the world. Buddha was promulgating his doctrines in India and Confucius and Lao-Tze were teaching in China. It is not known how much Pythagoras knew about other parts of the world; but he was a student of Thales and almost certainly spent some time in Egypt.

Later, Pythagoras founded a school in Crotona. All of the teaching was oral as was the custom then. He exalted arithmetic because his philosophy was based on the postulate that number is the cause of the various qualities of matter.

He stressed the mystic properties of numbers and considered arithmetic one of the four degrees of wisdom, the others being music, geometry, and spherics (astronomy).

Aristotle tells us that Pythagoras related the virtues to numbers, and Plutarch reports that he believed that the earth was produced from the regular hexahedron, fire from the pyramid, air from the octahedron, water from the icosahedron, and the heavenly sphere from the dodecahedron; in all of which the physical elements are related both to numbers and to form.

Philolaus was probably voicing teachings of the master when he asserted that 5 is the cause of color, 6 of cold, 7 of health, and 8 of love. The Chinese claimed that 5 represents wind, and 2 earth.

None of this, of course, has anything to do with the use of numbers in reckoning. This, in times of ancient Greek culture, was considered work and was left to the slaves. Most of it was done on counting boards, instruments similar to the abacus used in Japan and China today. Before the invention of the zero, it was impossible to write down the results of reckoning when there was no number on one column. If you have ever tried, or contemplated trying, a problem of multiplication while using Roman numerals, you will quickly understand (and approve) the wide use of counting boards.

The invention of a symbol for zero happened in India, sometime in the early centuries A.D. Some unknown Hindu put down a symbol, a dot he called *sunya*, to indicate a column in the counting board in which there was no bead. (It is interesting to note that the philosophy and religion of the Hindu peculiarly prepared him to invent a symbol for “nothing,” or the void.)

Notation reached Europe via the Arabs, as an Arabic notation, not immediately accepted despite its usefulness. In Arabic, the word was *sifr*, and the entire system of reckoning became identified by the name for that one important dot, or *cipher* (*zero* came much later from Italian). But *sifr* was not truly a number, only a symbol for an empty column.

The discovery of zero as a number came much later, perhaps centuries later. Mathematicians test historic works as to whether they are before or after the discovery of zero as a number by how zero is used in division. There are few mathematical complications about zero as a number in addition or subtraction, or even in multiplication. The problems develop mostly in division.

During at least one ancient era, there was a system of counting based on two. The system went: one, two, two plus one, two two’s, much. This is a most rudimentary system. Large numbers were not much used, but there was a way of saying, “plus the thumb” (one hand, or five, plus one of the other hand); or “to the pointer” (one hand plus the other as far as the pointing finger, or seven). The first part of this system is somewhat akin to computer calculations where everything is reduced to two digits, 0 and 1.

Thales (640-550 B.C.), the first of the seven sages of Greece, summed up the thought of the lifetime by saying, “Everything is water.” Today, some might say, “Everything is electricity.” Pythagoras (about 584-495 B.C.) might well have expressed his views by saying, “Everything is number!”

2.

NUMBERS may be considered in more than one way. Simply as numbers, much ingenious play is possible. For example: If any number is multiplied by 9; the digits resulting if added together, make 9. Take any figure, subtract its reverse, and the resulting digits add to 9. Take any number, subtract the sum of its digits, add the resulting digits and the result is 9.

A most curious number is 142,857. Multiply it by 1, 2, 3, 4, 5, or 6; the results give the same numbers in the same order, but beginning at a different point. Multiply it by 7, the result is all 9’s. Multiplying it by 8,yields 1,142,856; add the first and final digits, and you have the original number.

Another odd number is 37. Multiply it by 3, or any multiple of 3 up to 27, you will end up with three identical digits.

Or write down the cardinal numbers from 1 to 9, excluding 8. Multiply 9 by whichever number you wish to find in the final result; for example, 4 times 9 is 36. Now multiply 12,345,679 by 36, and the result is a row of 4’s.

Throughout history, numbers as symbols have been a part of legend and mystic wonders. Numbers have not only interested mathematicians, but have fascinated poets, philosophers, and priests. The changeless relationship of numbers to each other has seemed to point to some subtle significance. Almost every number has been given an esoteric meaning at one time or another, and for varying reasons.

The Egyptians considered 1 to be a sacred number because it is indivisible and yet it enters into all other numbers. It was the symbol of life, of mind, and of the creative spirit.

In the Pythagorean system, 3, was considered the perfect number, expressive of the beginning, middle, and end. Throughout the ages, 3 has been given greater prominence than any other number except perhaps 7. As the symbol of the Trinity it appears repeatedly in all parts of the Bible.

In classic mythology there were 3 Graces and 3 Furies; originally there were 3 Muses. Cerebus’s 3 heads, Neptune’s trident, the tripod of Delphi are also examples of the number 3. The number appears in fable, fiction, and history; it is often depicted in flags by the use of 3 colors. Consider it in nature: morning, noon, and night; fish, flesh, and fowl; water, ice, and snow; animal, vegetable, and mineral.

Because it is the first square, 4 was revered by the Pythagorans. They swore by it although their most holy number, the symbol of the absolute, was 10. This 10 resulted from adding 1 plus 2 plus 3 plus 4. They considered that 4 stood for equilibrium and for the earth because of the 4 (known) elements and the 4 cardinal points.

Solomon’s seal, the pentagram, was the symbol of the number 5, and this number was considered to be the number of dominion by knowledge. The Gnostic schools adopted it as their crest. It was much used in incantation and was often considered to be the symbol of man, who has 5 senses, 5 members (head and 4 limbs), 5 fingers, etc.

The symbol for 6 is two triangles, base to base. Six is called a perfect number and represents equilibrium and peace.

Seven is usually considered a lucky number. In olden times, because it was made up of 4 plus 3 (both of which are good numbers), 7 was always regarded as sacred and mystic and rivaled 3 in popularity. There are 7 days in the week, the body is supposed to replace itself every 7 years. Hypocrates divided the ages of man into 7, as later also did Shakespeare. The Egyptians knew of 7 planets, on which knowledge the 7-day week was based. But rather oddly, the ancient Peruvians also had a 7-day week although they did not know the planets.

The Pythagorean philosophers called 8 the number of justice because it divided equally into 4 plus 4, which divided equally into 2 plus 2, which divided equally into 1 plus 1. Also, because it was the first cube, it represented the cornerstone and capacity, therefore plenty.

Nine, being 3 times 3 (a good number), is also favorable. It represents 3 triangles, it means the equilibrium of the 3 worlds, and it is considered a good omen. The Chinese in particular reverenced this number. Some African tribes also salute their chief by this number of obeisances.

Even before the decimal system, 10 was considered a perfect number. Since we have 10 fingers and toes, the number was early important, as reckoning was done by them.

St. Augustine considered 11 evil, as a transgression of 10 (which is the number of the law). Thirteen has long been thought unlucky; 16 is lucky, being the square of a square; 19 was considered to be the number of the sun, hence of gold; and 28 implies the favor of moon which is an uncertain favor – so may be lucky or unlucky, depending. Fifty is a lucky number to the Cabalists, also 60.

Most of the sacred and beneficent numbers are odd, and gamblers usually feel that odd numbers are the luckier. Among ancient heathens even numbers were shunned because they were divisible by 2, a number Pythagoras and others denounced as a symbol of death and dissolution and evil augury generally.

Numerology is a fairly new word (appearing in dictionaries only since about 1935), but number-mysticism, the study of the occult significance of numbers, is an ancient art and, as numerology, is widely practiced today. Even those who do not believe in it often do believe they have a lucky number. I was once told that my lucky number is 2 – but I wouldn’t bet money on it.

When grandfather was a boy, most people made their clothing on spinning wheels. Now some people lose their shirts on them. They should neither blame nor rely on the numbers; the numbers themselves are innocent. Always, as in the past and doubtless into the future, numbers are fascinating.

**Colophon**

Hand set in Deepdene; display type is Normandic Contornata and Stymie Light. Text paper is Warren’s Olde Style. Edited and published by Jake Warner. When we acquired an SP-15 Vandercook proof press, Harold Segal said that if he had one he would print his journal on it. Not to be outdumbed by Harold, these 475 copies were so printed. It’s a lot of work. Variations in inking are probably due to inexperience with the press.

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Greenbelt, Maryland 20770