Astrology is a mathematical study of biological and psychological dynamics. It is the oldest known science, and all other sciences appear to have been its children. It comes to us out of prehistoric times, and suggests the existence of a prehistoric civilization enlightened far beyond our own in some respects. It was kept alive in the East, but it degenerated into much that was little more than superstition. It became associated with mysticism, fatalism and cultism, was taught only in secret schools, and was practically outlawed with the dawn of the Christian era.

Christianity taught a doctrine of free will. Fatalism was in conflict with this doctrine and since it was believed that astrology was fatalism, it was outlawed. (See "THE STAR CROSSED RENAISSANCE" by CAMERON ALLEN, DUKE UNIVERSITY PRESS). Although its funeral was held, its own dynamics would not allow it to die.

A new word was coined. It was astronomy, a new science to take the place of astrology. Originally, all astronomers were astrologers. The new science was to ultimately deny any association between Planets and Man, but this did not come about immediately. In the beginning, it was merely the desire to deny astrology insofar as it might conflict with the freewill doctrine of the church. In the early 16th century, Paraselsus defined the difference between astrology and astronomy of that time. He stated that astronomy confined itself to character delinations, while astrology was used to predict the future. He claimed that astrology could not predict for the individual because he had a choice that he could make that would affect the future, but he held that astrology could predict the future of the masses. In effect, he was arguing that Man can live above his animal nature.

Modern textbooks carefully conceal the fact that the greatest minds they glorify are those of the astrologers of history. It is regarded as "good taste" when textbook writers delete anything and everything that demonstrates the part astrology has played in history. Some of these astrologers have been Aristarchus, Pythagoras, Ptoleiny, Copernicus, Kepler, Newton, Paracelsus, Tycho Brahe, Benjamin Franklin and many others. Modern astronomers like to teach us that Copernicus discovered that the earth was not the center of the solar system, but Copernicus was merely going back to a defense of Aristarchus, who taught a heliocentric astrology 200 years before the birth of Christ, and who learned it from the work of Pythagoras who lived 500 B.C. While glorifying all these names, the astronomers try to claim them as astronomers when they were actually astrologers.

To study astrology, one has had to be an individualist, rugged enough to brush off the criticism of orthodox education. One could not study astrology and be a conformist.

Perhaps the three men who influenced the dawn of the 20th century astrology in a new direction were Allen Leo, Sepharial and Llewellyn George. They provided enough printed literature for new investigators to find a place to sink their teeth. Without them, it might have taken us another century to get started. The color and romanticism of Evangeline Adams probably drew more public attention.

In this course, astrology will not be taught as a science of cause and effect, but as an abstract science, a branch of mathematics. To add 6 and 10, you do not need apples. It will not be taught that planets affect us. We do not know that they do. We do know that we have a complicated realm of coincidence. It would appear that cause and effect operate in accord with certain abstract, mathematical laws which condition their functioning, and that astrology explains these laws.

Six apples and ten apples are sixteen apples. Six oranges and ten oranges are sixteen oranges. The oranges have nothing to do with the apples.

One does not cause the other, but the same arithmetical principles that apply to the apples apply also to the oranges. Mathematical principles are universal principles. They work anywhere, any time. Astrology deals with these same principles.

There have been many attempts to explain astrology in terms of physical science, vibrations, electricity or light waves, but they are too simple, while astrology is too complicated and too all-inclusive. Astrology seems to include everything. It involves the physical, the mental, the emotional, the vital, the concrete, the abstract. It even involves time and space, and strangely relates all of these things to each other. It is the mathematics of Life. (Paul G. Clancy called it the Algebra of Life.) It is a study of the abstract principles of existence, and there can be existence only in accord with those principles. Astrology explains life in terms of geometry.

It draws strange parallels that can never be found without it. What connection might there be, for example, between executive ability and the sex organs, between interest in religion and the liver? We do not know, but astrology indicates that there is. There are many other strange parallels. What you will learn here is merely what the writer has been able to dig out in 47 years. The principal thing he has learned in that time is that he hasn't scratched the surface. It is desirable that the student make no attempt to move too rapidly. There are certain basic factors that must be grasped. Speed may slow you down. Astrology is unlike anything else you may have studied. We want you to prove every point to your own satisfaction.

In the end, we want you to take our word for absolutely nothing. Astrology works, but it has to work exactly the same for you as it does for us. If we claim that 6 apples and 10 apples equal fifteen apples, while you find that 6 oranges and 10 oranges equal sixteen oranges, there is something wrong. We must find agreement all the way.

In the writer's approach to this subject, he had to deal with many erroneous claims. He had to discover which claims were true and which claims were false. He was told that you have to accept astrology on faith. This was the most fabulous claim of all. Accept absolutely nothing on faith. Astrology has to work in laboratory experiments. When you have finished this course, you will merely have reached a starting point. You will be like the salesman who was trained in the New York office and then sent to Mexico City to start selling. Certain principles that worked in New York will also work in Mexico City, but down there folks speak Spanish.

Suppose we could say that there are twelve basic material elements and that all other elements are combinations of these elements. It would simplify things. Astrology deals with twelve basic principles, and everything grows out of intricate combinations and relationships involving these twelve basic principles. The number of combinations may not be infinite, but it might as well be. It is very great. An astrologer is only as good as his understanding of these twelve basic principles. Our greatest objective must be to learn as much as possible about these twelve principles. You will not know all there is to know from us, because we do not know that much.

There is evidence that the ancients may have known that civilization was going into an eclipse, and that they tried very hard to put certain information, hidden away in symbol form, where it could ultimately be found, interpreted and revived. There are some who believe that the construction of the Great Pyramid was for the purpose of preserving mathematical and astrological knowledge during the dark ages until new initiates with an ability to understand abstract principles could decipher the information that is hidden away in the dimensions and measurements of the pyramid. The Mexican Calendar Stone was carefully buried, only to be found many centuries later and to reveal a calendar more perfect than any then known to modern science. The first Spanish explorers, arriving in a new and unknown land, found the symbols of the signs of the zodiac carved on stone throughout Mexico, Central and South America. (See "RED MAN'S CONTINENT" by ELLSWORTH HUNTINGTON, YALE UNIVERSITY PRESS.)

It would seem that all religious writers were attempting to get across the significance of two numbers -- 12 and 7. Twelve tribes of Israel, twelve disciples of Christ, the Book with the Seven Seals, etc. The Sphinx is an astrological symbol. Ancient religious works are filled with terms which have an entirely different meanings in astrological terminology. This includes the Bible. If you possessed information, it was necessary to hide it in order to avoid persecution. These facts need not seem strange when you realize that there are towns in the United States today which have laws forbidding anyone to teach astrology. The Federal Communications Commission uses its pressure to keep it from being taught on the air, and radio and television stations live in fear of having their license revoked if they disobey the wishes of those non-elected persons who constitute a Washington bureaucracy. As a result, astrological groups have sometimes incorporated as a religion, because under the constitution, you can't outlaw religion. This same procedure has been followed by advocates of Dianetics, who have changed their subject to Scientology and called it a religion to avoid persecution by the American Medical Association, an institution bent upon a monopoly of all things related to human health and even social relationships. The ultimate goal of socialized medicine would achieve such an aim, but it has been opposed by a great part of the medical profession because it might take control away from the A.M.A.

It is unlawful to ship astrological literature to Russia. It was a paradox to see an astrological convention opened in Los Angeles, where astrology has been outlawed, but openly practiced, by the man who was the Lieutenant Governor and later Governor. You can practice astrology anywhere in the United States if you call it a religion. We practice it throughout the United States, and we do not call it a religion. The Post Office Department has no objection to astrology You can practice astrology anywhere in the U.S., if you call it a study of cycles. While the FCC bars astrology by name, they daily utilize data furnished by the Radio Corporation of America predicting magnetic storms from a study of the planets. Although we point out these facts and inconsistencies, they should not bother you. We have been in business for 37 years, and they have neither bothered nor hindered us. It will take time and a great deal of money to eradicate ignorance from high places.

Astrology bases everything on the numeral 12, while mathematicians long claimed that the most perfect system of mathematics would be based on twelve instead of ten, the duodecimal system, instead of the decimal system. This fact appears to have significance. To understand the significance of the seven throughout ancient literature, it is only necessary to realize that in ancient mathematics, there was no zero. That was later invented by the Hindus. When a circle was cut in twelve parts, and the lines between the parts were numbered, there was no zero point. The first line was numbered as one instead of zero, and therefore, the halfway mark was seven, not six. Seven is not one-half of twelve, but you start counting the second half with seven, just as you start counting the first half with one. Astrology's seventh principle is a very dynamic one, as we will later see. It deals with time and change, for without change time ceases to exist. In the abstract world, there is no time. Time is merely an expression of matter and space. There is psychological time, but it is unlike clock time. (In recent years, many mathematicians have shifted their faith to the binary system of math, based on 2, but this is probably a fad, based on the fact that modern electronic calculators employ the binary system plus another system based on 16, because 16 is the 4th power of 2. The binary system was discovered by Leibnitz who lived 1646 to 1716 A.D. and who employed it to try to convert the Emperor of China to Christianity.)

Although the material parts of the solar system are the only material factors with which astrology deals, there are many other factors, but they are geometrical and in the realms of pure abstraction. It should be mentioned that there are schools of astrology attempting to deal with fixed stars, but we recommend that the student stay away from any such theories until he has finished Lesson Eleven of this course. After that, he is on his own. You will find that the astrologer speaks of things such as the signs of the zodiac as if they were as real as the Empire State Building. They are as real, but they are not material. They are mere Geometrical angles and mathematical points. In his early days of investigation the writer tried to visualize the zodiac as a possible magnetic field, but this conception breaks down, unless a magnetic field is something quite different from what we suppose it to be. Also, the poles of the earth's magnetic field do not conform to the poles of the zodiac.

The average astrology student of today has no clear conception of what the zodiac is. He may view it as something extending far out in space, but it isn't. It is all right here. It doesn't necessarily extend beyond the surface of the earth. It has no connection with the fixed stars. It is an abstraction. It is a group of mathematical points which can be found from the relationship of the earth's equator to the earth's orbit. It would have no significance for a being born on Mars or any other planet. It is purely a product of motion -- the earth's motion -- two earth motions, the rotation of the earth on its own axis and the revolution of the earth along its own orbit. It is geometry, involving two planes of space, the plane of the earth's equator and the plane of its orbit. To determine the zodiac, it is necessary to consider no other factors in space, except for the fact that there are two foci to the ellipse of the earth's orbit, and we must know which of these is occupied by the Sun.

It is important that you have a clear conception of what the zodiac is, and what it is not, for then you can be saved many arguments with yourself or others as to why certain Systems of astrology being practiced are without valid foundation. Attempts to use a heliocentric (Sun as center) zodiac fall down, because there is nothing about the zodiac that is heliocentric. We cannot draw a heliocentric zodiac, because we know too little about the path the Sun is taking through space. Each planet would have its own zodiac, which would have to be calculated from the relationship of its equator to its orbit and it would be quite different from our zodiac. Its twelve points would not be in the same place. The zodiac, as we use it, constitutes a series of dynamics. If either motion of the earth were to suddenly cease, the zodiac would also cease to exist. Remember that. The zodiac is a product of motion alone. When you realize this, you will not be harboring the illusions concerning it that are prevalent in many minds today. Everything in astrology concerns dynamics and kindred phenomena. Without motion, astrology would not exist.

Just as Euclid's geometry failed to meet the requirements of a mechanical and atomic age, some of the old conceptions of astrology must give way to the new geometry of dynamics. Euclid's geometry failed to consider dynamics. Einstein and Relativity have changed things. There is evidence that the arctic may have, at one time, been tropical, meaning that the equator was across a different part of the earth's surface At such a time, the zodiac may have been different from what it is today. We have a diagram that follows.

It will require some explanation. It is intentionally drawn out of proportion in order to illustrate and emphasize certain factors. You will see four large circles and one small circle. The small circle represents the Sun, which is actually many, many times the size of the earth. The four larger circles represent four positions of the earth in its trip around the Sun, spring, summer, autumn and winter. If you turn the diagram sideways, spring will be at the top, summer at the left, autumn at the bottom and winter at the right.. Note the odd fact that the earth is nearer to the Sun when we, in the Northern Hemisphere, have winter, farthest away from the Sun when we, in the Northern Hemisphere, have summer. The reverse would be true for people living south of the equator in the Southern Hemisphere. If heat is really something that travels through space from the Sun to the earth, then the average temperature south of the equator should be warmer in summer and colder in winter than summer and winters in the Northern Hemisphere. It is believed by some, however, that because there is far more ocean south of the equator than north of the equator, the ocean absorbs heat from the Sun in summer and gives that heat off in the winter, modifying temperatures south of the equator from what they might be otherwise. It does seem odd that when we, north of the equator, have winter, the earth is 3 million miles closer to the Sun than in summer. There is the alternative possibility that heat as we know it may be generated here at the earth AFTER light from the Sun reaches the earth, in which event, distance of a planet from the Sun might have no bearing on the temperature of that planet, a possibility that materialists, dogmatists and astronomers would hate to entertain.

This diagram will emphasize the fact that the earth travels in an ellipse, not in a circle and that it is closest to the Sun when we have winter. Now, we run into the difficulty of trying to illustrate a 3- dimensional factor on a. 2-dimensional piece of paper. Each circle representing the earth has a solid straight line representing the axis of the earth, with the north pole at the top, the south pole at the bottom. However, for the pole to be at this angle, you would have to be looking at the earth with your eyes in the plane of the earth's orbit. Suppose you hold the earth in the position given, but drop the spring position of the earth down so that it is behind the Sun, raising the autumn position so that it is in front of the Sun. You would no longer be able to see the Sun because it would be concealed by the autumn position of the earth, which would also conceal the spring position of the earth, which would be behind both the autumn position of the earth and the Sun. Then, you would find the poles of the earth exactly as given. The north pole would always tilt toward the right. It would not turn around as the pole of a spinning top would do. It would not tilt to the left when in its summer position The dotted straight line across the earth represents the equator. Note that in summer, the northern hemisphere tilts toward the Sun, while in winter it tilts away from the Sun. This angle at which the hemisphere tilts seems to be the most important factor in determining temperature.

However, there are only two moments out of a year when a straight line drawn from the Sun to the earth will also pass through the plane of the earth's equator. At those two moments and only during those two moments, can a straight line pass through both the plane of the earth's orbit and the plane of the equator. These two moments are known as the spring and autumnal equinoxes, and they become the markers upon which we can base the whole zodiac. Draw a straight line from the earth to the Focus occupied by the Sun at one of these moments, and allow it to point always in the same direction. The earth moves on. This line no longer points to the Sun, until six months later, the other end of the line will point to the Sun. The line will point at no specific point on the earth's surface, because the earth itself is rotating within its zodiac. Temporarily, and for a matter of years, the line will point at some fixed point in space, like a star, but even this is not permanent. The line points at absolutely nothing material permanently. It is purely an abstraction, and yet, upon this abstraction all else is based.

This could all be wild dreaming except for one fact. Conclusions based on this line function. They work. They can be depended upon. This brings up the great question. How did men of an unknown time in history figure all this out? It seems impossible that it could have come to light through observation. It must have been calculated by some master mathematician who knew something about human dynamics and mathematical principles that we do not know today. His discovery eclipses the discovery of the atom by modern scientists. Next, using this abstract line, he drew a circle around the earth in the plane of the earth's orbit, and divided it into twelve equal parts, and these became the signs of the zodiac. He knew something about twelve that we do not know. He may have been a student of the duodecimal system.

We have reached the point where we have a circle divided into twelve equal parts. Up till now, it is a mathematical figure -- no more, no less. The astrologer finds it related to twelve principles in nature, quite extraordinary principles. We are going to devote a great deal of time to these principles. In fact, you can devote the rest of your life to them, and you will always be learning something new. They are the principles of human dynamics. It would be truthful to say they are the principles of biological dynamics, for they do not confine themselves to humans. They apply for a dog or a cat, but we cannot study astrology without considering two other factors, evolution and heredity. As the writer has stated elsewhere, a horse can never become an accomplished pianist because of his heredity. Heredity is definitely a limiting factor. There is more fatalism to heredity than to astrology.

Throughout your study of astrology, this mathematical figure is going to be your ruler. You are going to use it for purposes of measurement. The zodiac is not the only place where you will use it. It will be used elsewhere. In fact it can be used in so many different places that it knocks any idea of a magnetic field into a cocked hat, unless there are millions and millions of magnetic fields all around us, and there may be, we do not know.

Mathematical rules work regardless of time or space. They are abstractions. They are, in no way, affected by circumstances. They do not change. Observations made by Euclid or Pythagorus 2500 years ago can be made today. Even the mountains and fixed stars have changed in certain respects during that time, but not the rules of mathematics.

We need an illustration that will furnish us with a parallel, and we are going to utilize a well-known mathematical formula for that illustration. Please don't run away. We will make it simple.

We have stated that astrology is a study of biological dynamics. Perhaps we should purify this even more. Although we use astrology to grasp and understand biological dynamics, there is a part of astrology which can only be classified as a higher branch of mathematics. Now, let us strike our parallel. We are going to use the Pythagorean Theorum.

Why not draw a right-angled triangle, meaning one where one of the angles is 90 degrees, like the corner of a square. Around this triangle, draw three squares. As a unit of measurement, use ¼ inch. The sides of this triangle are 3, 4 and 5 units. The longest line in this triangle is known as the hypotenuse. That is the one marked 5. Pythagorus discovered that in any such triangle, the area of the large square is exactly equal to the area of the two smaller squares. Ever since his time, this has been known as the Pythagorean Theorum. (However, it was actually known by the ancient Chinese earlier.) The size of the triangle, or the relative length of the two legs (short sides) of the triangle will never alter the rule. The Theorum is expressed: (a * a) + (b * b) equals (c * c).

To demonstrate the universality of the law, we ask you to draw a triangle, where the dimensions are 5, 12 and 13. You can make up millions of such triangles of all sizes, and the law will always work. In some cases, you will deal with fractions, but you can use any odd number for the short side and work out a triangle where all sides can be expressed in positive integers (whole numbers). To get the other two sides, all you need do is square the first side, and cut the result in half. Since the result is an odd number, and not divisible by two, you have to have an answer that is a positive integer (whole number) plus 0.5. Now add 0.5 to get the longest of the two sides, and deduct 0.5 to get the other side. In other words, if the short side of the triangle is 7, the square of 7 is 49, and when we cut 49 in two, we get 24.5. Add 0.5, and we get 25. Deduct 0.5, and we get 24. Thus (7 * 7) + (24 * 24) equals (25 * 25).

Even using all the odd numbers to infinity in this way will not give you all the possible triangles that can be made out of positive integers (whole numbers). There is an infinite number of others. You will also note that in this particular series of triangles, the sum of the two longer sides is always the square of the shorter side.

Having given this illustration, we have a very important point that we want to get across. In fact, this is vital to any real understanding of astrology. When all right-angled triangles conform to this law, we have coincidence. This does not mean that there is any cause-and-effect phenomenon involved. One triangle is not the cause, nor is another triangle the effect.

Just as this rule will work for all possible right-angled triangles, the division of a circle into twelve equal parts will work in astrology under many different unrelated conditions. Later, you will see that the right-angled triangle itself is very important in the study of astrology.

In mathematics, there are undoubtedly infinite other rules similar to the Pythagorean Theorum that can be applied to triangles in which no angle is 90 degrees. No mathematician has ever taken the time to work them out. Perhaps the idea has never entered his head. Never gain the impression that mathematicians have ever scratched more than the surface of their subject. There is so much about which we have never thought to inquire.

Everything, absolutely everything, in astrology is a study of angles and, mathematical points. We will even find that certain angles have some mysterious connection with the distance of an object from the central point of a sphere. The inference is that an object cannot move without setting in motion some very complicated system of dynamics. Time and space may assume entirely new meanings to you. The solar system is one monstrous atom, and Heaven knows what might happen if one of its electrons (planets) was pulled away. This is new geometry, but it is a dynamic geometry, not the dead geometry of Euclid.

We will save ourselves much trouble and many sleepless nights, if we start out by considering astrology, not as a cause-and-effect phenomenon, but as a system of higher mathematics. Do not let this scare or drive you away. It is important to your understanding, but there are vast numbers of students, using astrology from a practical point of view, who have almost no education whatever in the field of mathematics. One such student, doing very well otherwise, told the writer, "You know, I can't tell you how much is 9 X 9, without working it out in my own way."

Many students catch and grasp the basic principles with almost no knowledge whatever of the mathematical side of astrology. Yet, such people have helped to keep astrology alive down through the centuries. It must be stated, however, that under these circumstances, many errors, some of a mathematical nature, have crept into astrology as it has been taught over the last few hundred years. These must be brought to light and corrected. Under those circumstances, to think that the combined efforts of all its materialistic enemies have been unable to dent interest in the subject might almost be interpreted to mean that some guiding principles never intended to allow it to die out. This fact is as mysterious as the man who worked out these laws originally. It has been as diffIcult to wipe out astrology as to wipe out religion. There are some things that refuse to die. Astrology is one of them. It has outlived every enemy it ever had. It has outlived the civilizations of its enemies.

Thus, we have reached a point where we have a circle divided into twelve equal parts. Let us go on from there.

Because of the ancient's plan of hiding knowledge behind symbols many weird interpretations have sprung from their writings. We are going to take four ancient terms and tell you what they mean. The terms are FIRE, EARTH, AIR and WATER. Because of the constant use of these terms, which were astrological, moderns have convinced themselves that these ancients were under the impression that there are only four elements. In "Mathematics for the Million", Hogben points out that Pythagorus associated these words with triangles, and Hogben thought it was quite silly but amusing. So, let us tell you what these words mean. They are psychological terms.

- FIRE means VITAL
- EARTH means PHYSICAL
- AIR means INTELLECTUAL
- WATER means EMOTIONAL

The twelve signs of the zodiac were divided into four groups. There were FIRE signs, EARTH signs, AIR signs and WATER signs. The three signs of a group were always spaced 120 degrees apart, so that each group formed a perfect equilateral triangle, and the four groups, when fitted together formed the circle divided into twelve equal parts.

Do not lose sight of the fact that we are still in the realms of the abstract, because all this grows from that mathematical point in time, when it is possible to draw a straight line through both the plane of the earth's orbit and the plane of the earth's equator. We measure from there.

Biological dynamics have to do with a circle divided into twelve equal parts, representing the vital, the physical, the intellectual and the emotional -- four equilateral triangles, with three corners each, adding up to twelve, the base of the duodecimal system of mathematics. As we go along, you will find that the pieces all fit together perfectly, and everything always comes out even.

There were three more words--CARDINAL, FIXED and MUTABLE.

The zodiacal signs were also divided into four CARDINAL signs, four FIXED signs, and four MUTABLE signs, making up the twelve. This gave one Cardinal sign of each element, one Fixed sign of each element and one Mutable sign of each element.

Just as the zodiac can be divided into four equilateral triangles, it is also divided into three squares, one representing four Cardinal signs, one representing four Fixed signs, and one representing four Mutable signs.

We have expressed this in our diagrams with geometrical figures. Throughout astrology, you will always be dealing with geometrical figures. You will always be dealing with 12-sided figures or divisions thereof. This appears to be a basic principle of nature known to the ancients, but, let us point out that you can find this division of a circle into twelve parts elsewhere in nature. It is common to say that a snowflake has six sides, a pattern repeated six times. Actually, however, each one-sixth is the same pattern repeated twice, once mirrored. Therefore, like the zodiac every snowflake has TWELVE sides. Here again, we see the base of the duodecimal system.

To think of the zodiac and its implications as ridiculous is to think of a snowflake as riducu1ous. What is more ridiculous than that all snowflakes should have the same number of sides. Because we find that all snowflakes do have the same number of sides, we recognize that a principle of nature is at work, and we accept the result. Crystals also form with six sides, and various types of rock formations do the same thing. Nature functions mathematically. In the zodiac also, we find a principle of nature at work, and we have to accept the result.

Some writers have drawn attention to the symbol of the Christian religion -- the cross -- and they maintain it to be the astrological square. The Bible is filled with astrological terms and symbology. The symbol of the Jewish religion is the six-pointed star, two equi1atera1 triangles. Actually, there should be no conflict between astrology and religion. There is no conflict between astrology and any legitimate branch of science. There may be conflict between astrology and unproved theories and dogma but unlike astrology, scientific theories are often in the realm of fashions. They have no permanence. We need think only of the billiard ball conception of the atom that was prevalent not so long ago. The modern theory of the atom conforms more to the ancient's saying,"as above, so below." Because RCA set up their own investigative and research department, their Propagation Department, and dropped accepted and fashionable theories, they found that geometrical figures formed by the planets can be used to predict the condition of terrestrial magnetism at any and all times. The geometrical figures established by their research were the same established by the astrologers long before the dawn of any written history. The Astrological Alphabet There are 31 principle letters in the astrological alphabet. Ultimately you will have to memorize these, and we present them herewith.

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last modified on January 19, 2006

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