## Mind Over Mathematics: How Gauss Determined the Date of His Birth

By Bruce Director

This afternoon, I will introduce you to the mind of Carl Friedrich Gauss, the great 19th century German mathematical physicist, who, by all rights, would be revered by all Americans, if they knew him. Of course, in the short time allotted, we can only glimpse a corner of Gauss’s, great and productive mind, but even a small glimpse, into a creative genius, by working through a discovery of principle, gives you the opportunity to gain an insight into your own creative potential.

I caution you in advance, some concentration will be required over the next few minutes, in order to capture the germ of Gauss’s creative genius. So stick with me, and you will be greatly rewarded.

I have chosen to look into a subject, which most of you, and even your children, think you know something about: arithmetic. Plato, in the Seventh Book of the Republic, says that all political leaders must study this science, because arithmetic is the science whose true use is “simply to draw the soul towards being.” Arithmetic, Plato says, is never rightly used, and mostly studied by amateurs, like merchants or retail traders for the purpose of buying and selling. Instead, political leaders, must study arithmetic, “until they see the nature of numbers with the mind only, for their military use, and the use of the soul itself; and because this will be the easiest way for the soul to pass from becoming to truth and being.”

By now, you may already get the hint, that what Plato and Gauss, meant by arithmetic, is not, the buying and selling arithmetic, you and your children were taught in school. Gauss called his, the “higher arithmetic,” and in 1801, he published the definitive study into higher arithmetic, called by its Latin name, {Disquisitiones Arithmeticae.} As you will see, if your teachers had been interested in training true citizens of a republic, higher arithmetic, is what you would have learned; not the amateurish calculations, needed by fast-food cashiers, stock-brokers, derivatives-traders, or calculating methods of the statisticians and actuaries who determine what lives are cost-effective, for HMO’s and insurance companies.

Unfortunately, the Enlightenment dominates the thinking of most people today, comprising an Empire of the Mind, where people instinctively stick to simple addition, even though a higher ordering principle is discoverable. They labor under the illusion, that adding the numbers one-by-one, is their only choice, simply because the creative powers of their minds are unknown to them. It is the underlying assumptions, of which people are not even aware, which determine their view of the world. Only by becoming conscious of these underlying assumptions, and then changing them, can any scientific discovery be made.

Now let’s catch a further glimpse of Gauss’s genius, (and a little of our own) by turning to another, more profound application of higher arithmetic.

Humble Beginnings

Carl Friedrich Gauss, came from a very humble family. His father, Gebhard Dietrich Gauss was a bricklayer; his mother, Dorothea Benz, the daughter of a stonemason. She had no formal schooling, could not write, and could scarcely read. They were married April 25, 1776, three months before the signing of the American Declaration of Independence. Sometime in the Spring of the next year, Dorothea gave birth to Carl Friedrich.

Being barely literate, Gauss’s mother could not remember the date of her first son’s birth. All that she could remember, was that it was a Wednesday, eight days before Ascension Day, which occurs 40 days after Easter Sunday. This was not necessarily an unusual circumstance in those days, as most parents were preoccupied with keeping their infant children alive. Once the struggle for life was secured, the actual date of birth might have gone unrecorded.

Twenty-two years later, the mother’s lapse of memory, provoked the son to employ the principles of higher arithmetic, to measure astronomical phenomena, with a discovery grounded in the principle that the cognitive powers of the human mind are congruent with the ordering principles of the physical universe.

In 1799, Gauss determined the exact date of his birth to be April 30, 1777, by developing a method for calculating the date of Easter Sunday, for any year, past, present or future.

A lesser man, wanting to know such a bit of personal information, would have relied on an established authority, by looking it up in an old calendar, or some other table of astronomical events.

Not Gauss! He saw, in the riddle of his own birth-date, an opportunity, to bring into his mind, as a unified idea, the relationship of his own life, to the universe as a whole.

The date that Easter is celebrated, which changes from year to year, is related to three distinct astronomical events. Easter Sunday, falls on the first Sunday, following the first full moon (the Paschal Moon), following the first day of Spring, (the vernal equinox). Because of Easter’s spiritual significance, and its relationship to these astronomical phenomena, finding a general method for the precise calculation of the date of Easter, had long been a matter of scientific inquiry.

Let’s look at what’s involved in the problem of determining this date. You have the three astronomical events to account for. First, is the year, the interval from one vernal equinox to the next, which reflects the rotation of the earth around the sun. Second, the phases of the moon, from new moon to full moon to new again, which reflects the rotation of the moon around the earth; and third, is the calendar day, which reflects the rotation of the earth on its axis.

Of course, you can only “see” these astronomical events in your mind. No one has ever seen, with their eyes, the orbit of the earth around the sun, or the orbit of the moon around the earth. Not until modern space travel had anyone ever seen the rotation of the earth on its axis. We see with our eyes, the changes, in the phases of the moon, the changes in the position of the sun in the sky, and the change from day to night and back to day. We see “with our minds only”, the cause of that change. This type of knowledge is outside the Enlightenment’s straight jacket.

Each of these astronomical phenomena is an independent cycle of one rotation. The problem for calculation, is that, when compared with each other, these rotations do not form a perfect congruence (Fig. 1).

There are 365.2422 days in one year; 29.530 days in one rotation of the moon around the earth, called a synodic month; and 12.369 synodic months in one year. These figures are also only approximate, as the actual relations change from year to year, depending on other astronomical phenomena.

Here we confront something, which was known to Plato, and specifically identified by Nicholas of Cusa, in his {On Leonard Ignorance.} There exists no perfect equality in the created world. Perfect equality, exists only in God. But since, man is created in the image of God, through his creative reason, man can rise above this limitation, and see the world with his mind, ever less imperfectly, as God sees it.

As Leibniz says in the {Monadology,} man can reflect on God with his reason only; and “we recognize, what is limited in us, is limitless in Him.”

So if there is no equality in the created world, we need a different concept. Our mathematics must be concerned with some other relationship than equality, if we are to successfully measure the created world.

A New Type of Mathematics

Gauss did this by inventing an entirely new type of mathematics. A mathematics, which reflected the creative process of his own mind. If the mathematics accurately reflects the workings of the mind, it will accurately reflect the workings of the created world, as any Christian Platonist would know.

This is real mathematics, not the Enlightenment’s dead mathematics of Leonhard Euler and today’s illiterate computer nerds, like Bill Gates, who think a computer is the same as the human mind. Their mathematics is no more than a system of rules to be obeyed. The Enlightenment imposes a false separation between the spiritual and physical realms. If the physical world doesn’t conform to the mathematics, the Enlightenment decrees, there is something wrong with the physical world, not the mathematics! And, if the creative mind rebels against the dead mathematics of Euler and Gates? The Enlightenment demands that the mind {submit} to the tyranny of mathematics.

Gauss’s higher arithmetic begins with a concept different from simple equality. The concept of “congruence.” Here again, you see how you and your children have been lied to by your teachers. Most of you have been taught, that congruence is the same as equality, when applied to geometrical figures, such as equal triangles. Not true.

Gauss’s concept of congruence, follows the concept of congruence developed by Johannes Kepler, in the second book of the {Harmonies of the World.} The word congruence, Kepler says, means to Latin speakers, what harmonia, means to Greek speakers. In fact the word harmonia, and arithmetic, both come from the same Greek root. Instead of equality, congruence means harmonic relations.

Here are some examples of what Kepler means by congruence (Fig. 2). As you imagine, in the plane, I can increase the size and number of sides, of each polygon, without bound. But, when I try to fit polygons together, with one another, I bump into a boundary. Triangles, squares, and hexagons are perfectly congruent. Pentagons, for example, are not (Fig. 3). In some cases, when I mix polygons together, such as octagons and squares, I can make a mixed congruence.

However, when I go from two to three dimensions, and try to form a solid angle, the boundary conditions for congruence change. For example, pentagons, which aren’t congruent in a plane, are congruent in a solid angle (Fig. 4). Hexagons, which are congruent in a plane, are not congruent in a solid angle.

So you see, the type of congruences which can be formed from polygons, is dependent on the domain, in which the action is taking place.

Gauss carried this concept of congruence over into arithmetic, using whole numbers alone. Two whole numbers are said to be congruent, relative to a third whole number, if the difference between them is divisible by that third number. The third number is called the modulus (Fig. 5). Gauss designated the symbol {@id} to distinguish congruence from equality ({=}).

You may recognize a similarity between the concept of congruence with the idea of musical intervals. In higher arithmetic, it is the interval between two numbers, and relationship between those intervals, which concern us. Just as in music, it is the intervals, and the relationship between the intervals, which communicates the musical ideas, not the notes themselves.

Another property of congruent numbers, is that they leave the same remainder when divided by the modulus (Fig. 6). These remainders are called least positive residues. For example, 16 and 11 are both congruent to 1 modulo 5. In higher arithmetic, numbers are related, not by their equality, but by their similarity of difference, with respect to a given modulus.

There are other important relationships among congruent numbers. For example, if two numbers are congruent relative to a given modulus, they will be congruent to a modulus which divides that modulus. For example, if 1,997 is congruent to 1,941 modulo 28, they will also be congruent, relative to modulus 4 and modulus 7, as 4 X 7 = 28 (Fig. 7).

Here we are ordering the numbers, not according to their “natural” given order, but according to a mental concept of congruence. In this way, we make the numbers work for our mind, not enslave our minds, to the order of the numbers.

Calculating the date of Easter

For purposes of our present problem, calculating the date of Easter Sunday for any year, you can think of the astronomical cycle as the modulus. The day, the year, and the synodic month, are all different moduli. The scientific question to solve, is, how can these three moduli, be made congruent?

If this weren’t hard enough, we still have another problem: the imperfection of human knowledge. This reflects itself in the problem of the calendar.

In 45 B.C., Julius Caesar, decreed the use of a calendar throughout the Roman Empire, that approximated the length of the year as 365 and 1/4 days. The 1/4 day, was accounted for, by adding one day to the year, every fourth year, the familiar “leap year.” In the language of Gauss’s higher arithmetic, the years are in a cycle of congruences relative to modulus 4. Those years, which leave no remainder when divisible by 4, are leap years; those that leave a remainder of 1 are 1 year after a leap year, and so forth (Fig. 8).

However, as we have seen, the length of the year is not exactly 365 1/4 days. It’s a little bit shorter. This difference, is not very significant, in the span of one human life, but is significant over centuries, and millennia. In fact, the Julian calendar is off by {one day,} every 128 years. Such a difference may not concern you, if your mind is narrowly focused on your own physical existence. It {will} concern you, if you’re thinking of your own life with respect to posterity.

By 1582, the Julian calendar was off by ten days. The vernal equinox, the first day of Spring, was occurring on March 10th or 11th instead of March 21. Easter, therefore was also occurring earlier in the year. Both the material and spiritual world, had gotten out of whack.

So, in 1582, Pope Gregory XIII, put a new calendar into effect; ten days were dropped out of that year. In addition, the leap year skipped three out of four century years, and every fourth century year, would be a leap year; for example, the year 2000 will be a leap year, but 1900, 1800, and 1700, were not.

Thus, in order to calculate Easter Sunday, and thus determine his own birthday, Gauss had to make congruent, three astronomical phenomena, and two imperfect states of human knowledge!

He accomplished this by reference to two other cycles, or moduli. Because the synodic month and the calendar year, are unequal, the phases of the moon occur on different calendar days, from year to year. But every 19 years, the cycle repeats. So, for example, if the Paschal Moon occurs on say, March 23, in one year, it will occur on March 23, 19 years later. If the Paschal Moon occurs on April 11, the next year, it will occur on April 11, again in 19 years.

If we call the first year in this cycle “year 0,” the next year, “year 1,” the last year will be “year 18.” In this way, the calendar years in which the phases of the moon coincide, will be congruent to each other relative to modulus 19. So, if you divide the year by 19, those years with the same remainder, will have the same dates for the phases of the moon.

The calendar days on which the days of the week occur, also change from year to year. Today is Sunday, February 16. Next year February 16, will be on a Monday. Since there are seven days of the week, this cycle would repeat every seven years, but because every four years is a leap year, this cycle repeats itself, only every four x seven, or 28 years.

However, in the Gregorian calendar, this cycle is thrown off, by the century years. This cycle is called the solar cycle.

Gauss’s Algorithm

Prior to Gauss’s discovery, a complicated series of tables, was compiled from these cycles, by which one could determine the date of a specific astronomical occurrence. Gauss’s genius was to find a simple algorithm, by means of higher arithmetic, which didn’t require any tables, but simply the number of the year. I will illustrate it for you by example (Fig. 9)

Take the number of the year, divide by 19, call the remainder {a.} For 1997, a=2. In the language of higher arithmetic, 1997 is congruent to two, modulo 19. This tells you where, in the 19-year cycle of the phases of the moon, and the calendar day, the year 1997 falls.

Divide the year by four. Call the remainder {b.} For 1997, b=1. 1997 is congruent to one, modulo four. This tells you the relationship with the leap year cycle.

Divide the year by seven. Call the remainder {c.} For 1997, c=2. 1997 is congruent to two, modulo seven. This tells you the relationship between the calendar day, and the day of the week.

The next step is a little more complicated (Fig. 10): Divide (19a + M=24) by 30; call the remainder {d.} For 1997, d=2. This gives you the number of days, after the vernal equinox, that the Paschal Moon will appear. M changes from century to century, and is calculated from the cycle of dates on which the Paschal Moon occurs, in that century. For the 18th and 19th century, M=23. For the 20th century M=24.

Finally, divide (2b + 4c + 6d + N=5) by seven and call the remainder {e.} For 1997, e=6. This gives you the number of days from the Paschal Moon, to the next Sunday. This formula takes into account the relationship of the year to the solar cycle. N also changes from century to century and is based on the cycle of the days of the week on which the Paschal Moon occurs in that century. Sunday being 0, Monday being 1, Saturday being 6. For the twentieth century, N=5.

Gauss calculated the values of M and N into the 25th century, and derived a general method for calculating these values for any century. Unlike some people today, Gauss, was not planning on the “end times.”

Therefore, Easter Sunday is March 22 + d + e. For 1997 that is March 22 + 2 + 6 or March 30, 1997 (Fig. 11).

Gauss’s method, obviously has applications, far beyond the determination of his birthday, or the date of Easter Sunday, for any year. In his later work, Gauss brought even more complex astronomical observations into congruence, by use of these same powers of the mind. But, this little example gives you a sense of how a universal creative mind can take any problem, and see in it an opportunity to extend human knowledge beyond all previous bounds.

Of course, we too can learn a lesson from this. The next time a child asks you a question about how the world works, something like, “why does the moon change from day to day?” or, “why does the sun change its place during the day and over the course of the year?,” don’t tell that child to look up the answer in a book, or log onto the Internet. Help that child to discover how, as Plato says, to see the nature of numbers with the mind only.

Then, take that child, with this newly acquired discovery, outside and show him the night sky. Then, that child will be able to see, in that night sky, the image of the workings of his or own mind, and to see also, reflected back, in that image, an imperfect, yet faithful, image of the Creator, Himself.

## Measurement and Divisibility

By Bruce Director

In 1818, Karl F. Gauss accepted the assignment to conduct a geodesic survey of a large part of the Kingdom of Hannover, or, in other words, to measure a section of the surface of the Earth. The project involved many difficulties, and requires, first, that one reflect on the general concept of measurement.

Gauss’ friend and collaborator, the astronomer Bessel, thought a man with Gauss’ mathematical ability, should not be involved in such a practical project, to which Gauss replied:

“All the measurements in the world are not worth {one} theorem by which the science of eternal truths is genuinely advanced. However, you are not to judge on the absolute, but rather on the relative value. Such a value is without doubt possessed by the measurements by which my triangle system is to be connected with that of Krayenhoff, and thereby with the French and the English. However low you estimate this work, in my eyes it is higher than those occupations which are interrupted by it. … you will agree with me, that, when one does without all real help in numerous petty affairs, the feeling of losing one’s time can only be removed when one is conscious of pursuing a {great important} purpose…

“What do I have for such work, on which I myself, could place a higher value, except {fleeting hours of leisure?…”

How can you measure the surface of the Earth? Don’t even think about using a yardstick. First think what it means to measure. You cannot measure one thing by another, unless you first can determine, if the two things are commensurable. If you worked through the last several weeks’ pedagogical discussions, you know it is not always self-evident, whether two magnitudes are commensurable with each other.

To get a sense of this, look at a similar problem, investigated by Euclid, Archimedes, Cusa and Kepler, about which much commentary has already been written: Measuring the circumference of the circle.

One can measure a circle by another circle, or a part of a circle, but not by a line, or any other curve. A whole circle can measure another whole circle, only with respect to size, i.e., one circle is either greater or less than the circle by which it is measured. But, to measure along the circumference of the circle, the circle must be divided. The circumference can then be measured by the divided parts.

The first and most obvious division, is by half. This creates two semi-circles and a straight line diameter. Archimedes thought, that, by dividing the diameter into small parts, one could measure the circumference of the circle, but, Cusa proved, [and if you worked through last week’s pedagogical, you would have proved to yourself], that the diameter and circle are incommensurable. One cannot measure the other. So in order to measure the circle, we must divide the circumference itself into smaller parts.

Well, if we continue folding the circle in half and in half again, we will divide the circumference into smaller and smaller parts. The number of parts, will be powers of 2. (That is, 2, 4, 8, 16, ….) But other types of divisions must be discovered, if we want to measure a part of the circumference which is not a power of two.

If we unfold the circle, after folding it into quarters, we will have constructed, two diameters, which meet at the center of the circle. Now fold the circle, so a point on the circumference touches the center. This will form a new line, shorter than the diameter, which intersects the circumference in two points. Once this fold is made, it is easy to find two other folds which will also meet at the center, forming two more lines, which will make a triangle. (It is easier for you to discover this by experiment, than for me to describe it without the use of diagrams.) This divides the circle in three parts.

By a more complicated process, the circumference of the circle can be divided into five parts, the description of which, would require a digression here, but will be discussed in future briefings.

It was long assumed, and Kepler proved, that it were impossible to divide the circle into seven parts. Until Gauss, it was believed, that this was the ultimate boundary of the divisibility of the circumference of the circle. Gauss discovered the divisibility of the circle into 17 parts, and other divisions also. But for purposes of today’s discussion, what is important, is, that the process of division has a boundary. Not all divisions are possible, and since division is necessary for measurement, to measure requires one to discover, and if possible, overcome these boundaries.

To conduct his geodesic survey, Gauss had to determine how to divide the surface of the Earth, which presented many similar problems, albeit more complex, to our above example. For example, instead of measuring a curve, Gauss had to measure an area. This area, was on a curved surface, which in first approximation is a sphere, but is actually closer to an ellipsoid. How are these surfaces divided? How are these divisions, once discovered, measured on the surface of the earth itself? These and other problems, will be discussed in future pedagogicals.

But, while contemplating the above, it is not unhelpful to reflect on the following statement of Gauss, excerpted from his “Astronomical Inaugural Lecture” in which Gauss argues against the idea of sperating so-called practical, from so-called theoretical science:

“To judge in this way demonstrates not only how poor we are, but also how small, narrow, and indolent our minds are; it shows a disposition always to calculate the payoff before the work, a cold heart and a lack of feeling for everything that is great and honors man. One can unfortunately not deny that such a mode of thinking is not uncommon in our age, and I am convinced that this is closely connected with the catastrophes which have befallen many countries in recent times; do not mistake me, I don not talk of the general lack of concern for science, but of the source from which all this has come, of the tendency to everywhere look out for one’s advantage and to relate everything to one’s physical well-being, of the indifference towards great ideas, of the aversion to any effort which derives from pure enthusiasm: I believe that such attitudes, if they prevail, can be decisive in catastrophes of the kind we have experienced.”

Measurement and Divisibility Part II

Last week, we investigated the measurement of the circumfrence of the circle. What was required, was to divide the circumference into commensurable parts. It was demonstrated, that division by 2, and powers of 2, was possible by repeated folding and division by 3 was possible, by folding in a different way. Division by 5 was stated as possible, and left to the reader to accomplish, and division by 7 was stated to be impossible, and the reader was refered to Kepler’s proof (Harmony of the World, Book 1). To the eye, the circumference of the circle appears smooth, and everywhere the same, yet when one tries to divide the circle, one discovers boundaries, with each new {type} of division. Thus, the numbers 2, 3, 5, and 7 each signify a {type} of divisibility with respect to the circumference of the circle.

The word {type} here is used in the sense of Cantor and LaRouche. Each {type} of division, is seperated from the other, by a discontinuity. One cannot divide the circle into 3 parts, from the method of division by 2 or powers of 2. One can combine division by 2 and 3 to divide the circle into 6 parts, but a new {type} of division is required for 5 parts.

Let’s experiment with other types of divisions, with respect to other types of curves and surfaces.

Once the circle is divided, polygons can be formed by connecting the points on the circumference, with each other, and triangles can be formed, by connecting the vertices of the polygon, to the center of the circle. It is easily demonstrated, that these triangles are all equal. Thus, the relationship of all parts of the circumference to the center are the same.

Now look at an ellipse. The ellipse differs from the circle, in that all parts of the circumference of the ellipse have a relationship to two points, (called foci) not one, as in the case of the circle. Specifically, the distance from one focus, to the circumference of the ellipse, plus the distance from the circumference to the other focus is always the same. In the case where these two foci come together, and become one, the ellipse becomes a circle.

Look further at the ellipse. One can fold the ellipse in half in only two ways (which for convenience we can call horizontal and vertical), whereas, the circle can be folded in half in an infinite number of ways. When the ellipse is folded in half, one of the lines generated, will be longer than the other, the intersection of these two lines, (called axes) will be called the center of the ellipse. Two circles can be drawn, using this center, related to this ellipse. One will have the smaller line as its diameter, and the other will have the longer line as its diameter. The former will be smaller than the ellipse, the latter will be larger.

Now divide the larger circle into any possible number of parts, and form the triangles associated with the polygon which is formed by the division. The sides of the triangles, which correspond to radii of the circle, will intersect the circumference of the ellipse, dividing the circumference of the ellipse. Now connect the points of intersection with the circumference of the ellipse, to one another, forming triangles in the ellipse. It is easily seen, that unlike the circle, these triangles are not equal, consequently, the divisions of the circumference of the ellipse, formed by these divisions of the circle, are not equal. Hence, the ellipse, cannot be divided, or measured, in the same way as the circle. A new discontinuity has been reached.

This new discontinuity arises from the difference in the characteristic curvature, between the circle and the ellipse. The curvature of the circle is constant, while the curvature of the ellipse is always changing.

This problem, of measuring the circumference of the ellipse, a crucial problem for physics and astronomy, was investigated by Kepler, and further developed by Gauss, by applying his hypothesis of the complex domain. These issues will be investigated in future pedagogical discussions. But for now, take one more step. Now think of a sphere. By what method, can one divide the sphere in half, and what will this tell us about the underlying hypothesis concerning the divisions of the circle and the ellipse?

More next week.

MEASUREMENT AND DIVISIBILITY PART III

Last week’s discussion ended with the question: By what method can we divide a sphere in half? Let’s compare this problem, with the problem of dividing the circle in half. This was accomplished by folding the circle on itself, and, we discovered certain boundary conditions, with respect to that process. How can we apply this method to the problem of dividing a sphere?

First think about what we did when we folded the circle. We weren’t simply dividing the circle. We were applying a rotation to the circle, in a direction different then the rotation which generated the circle itself. That is, a circle of 2 dimensions, is rotated in 2 + 1 dimensions. Division in n dimensions, was effected by a transformation in n + 1 dimensions.

Now apply this to the sphere. Obviously the sphere can not be folded, but it can be spun. Or, in other words, if we consider the sphere, as a surface of 2 dimensions, we must take action in 2 + 1 dimensions, in order to divide it. So, if we pick a point on the surface of the sphere, and, spin the sphere around that point, every point on the sphere, except the one exactly opposite the initial point, will move. These two points can be connected by the equivalent of the diameter of the circle, which on the sphere is a great circle, that divides the sphere in half.

Now apply this principle, of measuring n dimensions, with respect to n+1 dimensions, to the initial discussion three weeks ago about Gauss’ efforts at measuring the surface of the Earth. How do we locate our initial position? With respect to north and south, we can measure the angle at which we observe the North Star. The higher overhead the North Star is, the farther north our position on the Earth. To measure our position on the surface of the Earth, we must look up, to the stars. This measurement is, therefore, n+1 dimensions, with respect to the n dimensions of the surface of the Earth. Now for our position with respect to east west, we must refer to the rotation of the earth on its axis, which goes from east to west. We measure this, with respect not only to a change in position with respect to heavenly bodies, but with respect to a change in time. Another dimension, (n+1)+1.

Once this position is determined, we now measure other locations in a similar manner, and then measure the distance between those locations, using triangles. In order to meaningfully measure the surface of the Earth, these triangles must be large. Too large to measure with rulers, yardsticks, or chains. If we start with two relatively close points on the earth, and precisely mark off the distance between them, we can then measure the distance between these two points and a third point, by measuring the angles that form the triangle between these three points. This is done, by placing an object at each point, that can be seen, using a telescope, from the other points, and we measure the angle at which the telescope has to be turned, to see each point.

Gauss invented a device, called the heliotrope, that used a small mirror to reflect sunlight, that could be seen, by a telescope, from many miles away. If three such devices are positioned at three different points on the Earth’s surface, a very large triangle can be formed, that can be measured precisely. In this way, the surface of the Earth, can be covered with a network of triangles, and measured.

But, when we look through these telescopes, to see each point, the light is refracted (bent), by the atmosphere, and the lens of the telescope. This makes what we see, different from the actual position of the point on the Earth. So this physical property, refraction of light, must be taken into account in our measurement–another dimension, [(n+1)+1]+1.

But since our measuring points are at different elevations, we use a level, which adjusts its position with respect to gravity. So we must measure variations of the gravitational field of the Earth, yet another dimension, {[(n+1)+1]+1}.

Likewise, when using a compass, which reflects changes in the magnetic field of the Earth, we must measure variations in the magnetic field of the Earth–yet another dimension, {[(n+1)+1]+1}+1. And so on, with each new physical principle discovered.

The inclusion of each new dimension is not a simple addition, but a transformation in the hypotheses underlying our conception of physical space-time. Just as the idea of dividing a circle, contained within it, an underlying assumption of a higher dimension, which wasn’t apparent, until thought of in terms of dividing the sphere, each new dimension, corresponding to a physical principle, uncovers previously “unseen” assumptions, with respect to the hypothesis of lower dimensions.

But, these assumptions, expressed in the form of anomalies and paradoxes, won’t be “seen,” unless you look for them, not in n dimensions, but in n+1 dimensions. You can’t measure where you are, except with respect to the horizon, which cannot be “seen”, except with respect to the higher dimensionality, which you are seeking to discover, but which you will not find, unless you have the passion to “look” for it.

## The Importance of Good Maps

by Bruce Director

As the pedagogical series on spherical geometry has indicated, a profound discovery arises, when you attempt to map spherical action on to a flat plane. Any such effort, immediately presents to the mind, the existence of two distinct types of action. Basic investigations of the physical universe, astronomy and geodesy, immediately confront us with the need to discover the conceptions that underlay this discontinuity.

Already we have presented several examples of this, which you can work through quickly in your mind before proceeding. Think of the various examples that demonstrated that spherical nature of the manifold of measurement of space. Think of the conception of the Platonic Solids from the standpoint of Kepler’s re-discovery of the Pythagorean concept of congruence (harmonia). Think how we demonstrated that solids arise as the characteristic perfect congruences on a surface of constant positive curvature, as distinct from the perfect congruences that arise on a surface of zero curvature. And also, think of the pentagramma mirificum, and emergences of two distinct periodicities that arise from carrying out the same action, on surfaces of two different curvatures. (All the above examples were elaborated in pedagogical discussions published over the first three months of 1999.)

Now let’s delve into this area once again. First, from the standpoint of mapping the stars, as represented on a surface of constant positive curvature, onto a surface of zero curvature, a most ancient investigation.

In our observation of the heavens, the stars are projected onto a spherical surface, as a function of our measuring their changing positions, as a change in the angle between the line of sight, the horizon and some arbitrary direction perpendicular to the horizon, such as north, or even “straight ahead.” In this way, the changes in position of the stars, and their relationship to each other, are represented as arcs of circles and the angles between such arcs.

However, as we’ve seen before, when we try to project this spherical projection of the stars, onto a flat surface, discontinuities aries. Furthermore, the nature of these discontinuities changes depending on how we effect that projection. In other words, not all projections from a sphere onto a plane are the same.

You can carry out a simple demonstration of this, by drawing a series of great circle arcs, intersecting at different angles, on a clear plastic hemisphere. (For purposes of this description, call the circular edge of the hemisphere the equator, and the pole of this equator the north pole.) Hold a flashlight or candle at the position equivalent to the south pole of the sphere so that the great circle arcs cast shadows onto a marker board. Trace the shadows. Now, move the flashlight toward the center of the sphere, stopping at various intervals, and tracing the shadows of the arcs at each interval. Make one of those intervals the center of the sphere. Trace the shadows.

You will notice a change in the curvature of the shadows, as the point of projection changes from the south pole to the center of the sphere. At the south pole of the sphere, the shadows are arcs of circles. As the flashlight moves toward the center, the shadows straighten out, until at the center, the shadows are straight lines.

Now make a more precise demonstration. Draw on the hemisphere, an equilateral spherical triangle, such as the face of the octahedron, that has three 90 degree angles. Perform the above projections. When the flashlight is at the south pole, trace the shadows. Now move the flashlight to the center of the sphere, and trace the shadows.

The tracings of the shadows from the south pole projection are circular arcs. Measure the angle between the lines tangent to each arc at the each vertex. Now measure the angles between the sides of the straight line shadows projected from the center.

These are two specific projections, the first called the stereographic, the second called central projection, that transforms the great circle arcs on the sphere, to the plane. As you can see, each transformation is different. In the central projection, the spherical equilateral triangle with three 90 degree angles is transformed into a flat equilateral triangle with three 60 degree angles. In the stereographic projection, the spherical triangle is transformed into three circular arcs that intersect each other in 90 degrees. So the angular relationship between the vertices of the triangle is invariant under the stereographic projection.

With a little bit of thought, you should be able to figure out why that is the case. Think of the point of projection as the apex of a cone of light. The projection on the flat surface is formed by the intersection of a line that starts at the point of projection, and continues through a point on the sphere, and then intersects the marker board. If the point of projection is at the center of the sphere, than the lines connecting the point of projection to points on a great circle, will all be in the same plane. Consequently, the projection of these great circle arcs will be a straight line. In this way, the center of the sphere can be thought of as the unique singularity from which great circles can be projected into straight lines!

Not so if the point of projection is other than the center of the sphere. However, if the point of projection is the south pole, the angles between the projected arcs, are the same as the angles between the spherical arcs. This property has come to be called, “conformal”.

Because of this angle preserving characteristic, this projection is particularly useful for mapping stars. The written discovery of the stereographic projection is attributed to the Greek astronomer Hipparchus, but its actual origins are most likely quite older. Under this projection, the entirety of the celestial sphere can be mapped onto a flat surface.

To do this, think of a sphere with a plane representing the horizon, going through the center of the sphere. (You can represent a cross section of this on a flat piece of paper as a circle with two perpendicular diameters. Call the endpoints of one of the diameters the north and south pole. Let the other diameter represent the horizon.) Now, draw a line that connects every point of the “northern” hemisphere with the south pole. Those lines will intersect the horizon and those intersections will form a stereographic projection. The north pole will project onto the center of the sphere. All the points of the northern hemisphere will project onto the inside of a circle formed by the intersection of the sphere with the plane, and all the points of the southern hemisphere will project to points outside that circle. Where will the south pole projet to? What other discontinuities or distortions emerge under this transformation?

YOU HAVE TO CARRY OUT THIS CONSTRUCTION IF THE ABOVE DESCRIPTION IS TO MAKE ANY SENSE TO YOU.

Over the last two millennia, the stereographic projection has been used to map the celestial sphere onto a plane and is the basis of the construction of the astrolabe, one of the earliest astronomical measuring instruments. (Rick Sanders has produced an interesting unpublished paper on the astrolabe available to those who are interested from RSS.)

The stereographic projection, therefore, represents a unique way of projecting one surface onto another, such that a certain characteristic, is invariant under the transformation. But, this projection is specific to the mapping of a sphere onto a plane. Can we find, for example in the case of a geodetic survey, where we are mapping the geoid, onto an ellipsoid, onto a sphere, onto a flat plane, a way to perform such a series of transformation, in which a certain characteristic, remains invariant under repeated arbitrary projections?

This formed the subject of Gauss’ famous 1822 paper for which he won the Copenhagen prize. The paper was titled, “General Solutions of he Problem to so Represent the Parts of One Given Surface upon another Given Surface that the Representation shall be Similar, in its Smallest Parts, to the Surface Represented.” In this investigation, Gauss delved even further into the nature of non-linear curvature in the infinitesimally small.

The Importance of Good Maps-Part II

Last week we undertook a preliminary investigation into the projection of a sphere onto a plane. Now the fun starts.

If you carried out the constructions, you would have re- discovered, in a formal sense, certain principles whose ancient discovery was crucial for the development of human civilization. That discovery can be thought of in two aspects; 1) that elementary form of action in the physical universe is curved, and 2) that curved action is of a different “transcendental cardinality” than linear action. The nature of that difference is revealed in the investigation, not simply of each type of action, but by investigating transformations between each type, i.e., the “in betweenness.” In that sense, the study of these projections has a significance for both the development of the higher cognitive powers of the mind, and the capacity of those powers to bring the physical universe increasingly under its dominion.

In general, there is no transformation of a sphere onto a plane that does not result in distortions and discontinuities, and it is by those distortions and discontinuities that the difference in “transcendental cardinalities” becomes apparent. But, there are a myriad of such transformations, each of which produces different characteristic distortions and discontinuities. (Last week, we investigated, preliminarily, two such transformations, the gnomic and the stereographic projection, but there are many others.) In order to more fully grasp the nature of the difference in “transcendental cardinalities” between the sphere and the plane, we cannot focus simply on specific types of transformations. We must investigate the general nature of transformations and not just between two specific types of surfaces, such as a sphere and a plane, but between any series of arbitrarily curved surfaces. That is, we must jump from investigating a particular projection, to the investigation of the general principle of projection itself. That puts us in the domain the hypergeometric. This is the domain unique to the contributions of Gauss and the subsequent discoveries of Riemann.

Today’s pedagogical discussion seeks to start down the road to the re-discovery of Gauss’ and Riemann’s contributions. There is nothing contained below that is beyond the scope of most of the readers, but, be prepared to concentrate on the train of thought. You will find in it an illustration, typical of Gauss, of taking a previously discovered principle of classical Greek science, and approaching it from a new higher standpoint, which establishes that classical principle, as a special case of a more general concept. It is congruent with Beethoven’s re-thinking of the significance of the Lydian interval, in his late quartets, to establish a new conceptualization of the domain of J.S. Bach’s well-tempered system of bel canto polyphony.

From last week’s discussion, you should have already demonstrated to yourself, some of the characteristics of the gnomonic and stereographic projection of the sphere onto the plane. Specifically, the gnomonic, (projection from the center of the sphere), transforms great circle arcs on the sphere, into straight lines on the plane. Obviously, since the sum of the angles of all plane triangles is 180 degrees, and the sum of the angles of triangles on the sphere are always greater than 180 degrees, angular relationships are changed under the gnomonic projection. On the other hand, last week’s constructions provided the basis to demonstrate, at least initially, that under the stereographic projection, i.e., where the point of projection is a pole of the sphere instead of the center, the angular relationships are unchanged when projected from the sphere onto the plane. This characteristic is obviously crucial for geodesy and astronomy, as the relationships between stars projected onto the celestial sphere and positions on the surface of the Earth, as these relationships are measured as only as angular relationships. If a representation of these spherical relationships onto a flat surface is to be of any use, the angular relationships must be invariant under the projection.

When thinking of possible projections from the sphere onto a plane, the gnomonic projection seems to suggest itself most easily. For example, in the case of the celestial sphere, the point of projection is the observer, who projects the celestial sphere the stars along the lines of sight from the observer through the stars, to a plane. This projection was apparently discovered by Thales, but it is quite possible that it was known much earlier. However, because it distorts angles, it has obvious failings for a useful map of the stars or the Earth.

The stereographic projection is much less obvious. Here, the point of projection, a pole, is no where in the manifold of the observer. But, when the projection plane is the plane of the observer, (as in last week’s example), the point of the observer is the only point that is unchanged under projection! This and the property that angular relationships are not changed under the projection, make the stereographic projection suitable for astronomical uses, such as a star chart, or astrolabe.

The experiment in last week’s discussion, for pedagogical purposes, indicated by demonstration, but did not prove, that angular relationships are invariant under the stereographic projection, a characteristic called “conformal.” One can, as Hipparchus did, prove by principles of Euclidean geometry, that this is the case.

(Such a proof is not very complicated. It relies on properties of similar triangles. But, to describe it in this cumbersome format would be, for the moment, distracting. So, we leave it to the reader to carry out.)

Gauss’ standpoint was to go beyond the principles of Euclidean geometry, by inverting the question. Instead of starting with stereographic projection and asking, “Is it conformal?” Gauss asked. “What is the nature of the being conformal, and under what projections does it exist?” The former sets out to discover the existence of a general principle in a specific case. The latter question seeks the nature of the general principle, under which the special cases are ordered.

Gauss’ approach is best grasped pedagogically by a demonstration. Take the clear plastic hemisphere you used last week, preferably with the 270 degree equilateral spherical triangle still draw on it. Cut out four circles out of cardboard, of different sizes. For my experiment, I made a circles with diameters, 3 1/2, 1 1/2, 1, 1/2. (For the circle of 1/2 inch diameter I used a thumb tack.) With tape, attach these circles to the sphere, all at the same “latitude”, so that they are approximately tangent to the sphere at their centers.

Now, project this arrangement onto a plane. This is most easily done, by holding the hemisphere so that the plane of the equator is parallel to a wall or the ceiling, and use a flashlight to project the spherical images onto the wall or ceiling.

When you hold the flashlight so that the bulb is at the center of the hemisphere, the shadows of the spherical triangle will, as we saw last week, be straight lines. The shadows of the tangent disks, will be ellipses. When you pull the flashlight back to the position of where the south pole of the sphere would be, you will see that the shadows of the spherical triangle will be circular arcs, intersecting at 90 degree angles, and the shadows of the tangent disks will be almost circular.

The change in the projection of the tangent disks, from ellipses in the gnomonic projection, to circles in the stereographic, is a reflection of a crucial element of Gauss’ discovery.

Gauss’ first step, was to abandon the idea of the sphere and plane being objects embedded in three dimensional Euclidean space, and instead, he thought of each as a two dimensional surface of different curvatures. On any two dimensional surface, the angular relationship of 90 degrees is a singularity, consistent with Cusa’s notion of maximum and minimum. That is, geodetic arcs, or lines that intersect at 90 degrees are at the maximum point of divergence. Or, in other words, any two such arcs, or lines, define two divergent directions. Any other angle, at which geodetic arcs lines intersect, is merely a combination of these two directions. (Gauss goes to great lengths to point out that these two directions are arbitrary, but once one is chosen, the other is determined.)

Now look back to the difference in the transformation of the tangent disks in the two projections. In the gnomonic projection, the change of those disks from circles to ellipses, is a reflection that the gnomonic projection changes one direction in a different way than the other. The transformation of those disks into circles in the stereographic projection, is a reflection of how this projection changes both directions exactly the same.

But, there is another principle at work here that you can discover with some careful observation. If you look closely at the tangent disks, you should notice that in the gnomonic projection, the shadows of the disks become more elliptical, the smaller the disk. And, in the stereographic projection, the shadows of the disks become more circular the smaller they are.

Remember these disks are not on the sphere, but tangent to it. Therefore, the smaller the disk, the closer to the surface of the sphere it is. As the disks become infinitesimally small, the characteristic change in curvature, becomes even more pronounced. In other words, the characteristic curvature of these projections, or any other for that matter, is reflected in every infinitesimally small area of both surfaces. And, the smaller the area, the more true is the reflection! Just the opposite of linearity in the small.

Do this experiment and play with this idea a while. You are getting close to a very fundamental principle discovered by Gauss and Riemann, which we’ll take up in the final installment of this series next week.

The Importance of Good Maps–Part 3

I hope you had fun conducting the experiment described at the end of the last pedagogical discussion. This week, we will conclude this preliminary phase of pedagogical discussions on the early development of the Gauss-Riemann theory of manifolds, with a discussion of the general principles of Gauss’ theory of conformal mapping. In future weeks, we can extend these investigations, using this preliminary work as a starting point.

It is important to remember the context in which these investigations of Gauss and Riemann occurred. The thread begins with Cusa’s {Learned Ignorance}, and his insistence that action in the physical universe was elementarily non-uniform. The discoveries of Kepler on planetary orbits, and Leibniz and Huygens on dynamics, and light, confirmed and validated what Cusa had anticipated. In each case, the general nature of the non- uniformity of physical action, was discovered by the manifestation of that characteristic in an infinitesimally small interval of action.

Gauss’ geodesy is a good case in point. Between 1821 and 1827 Gauss supervised and conducted a geodetic triangulation of most of the Kingdom of Hannover. That undertaking confronted him with a myriad of scientific problems, that sparked a series of fundamental discoveries about the nature of man and the physical universe.

A short review is necessary, from the standpoint of the last several month’s pedagogical discussions on spherical action. Think back to the question of the measurement of the positions of the stars with respect to a position on the Earth. Those positions will change over the course of the night, the course of the year, and the course of the longer equinoctial cycle. The geometrical form of the manifold of such changes, is the inside of sphere. The daily, yearly and equinoctial changes of the stars’ position trace curves on the inside of the sphere. Those curves can be thought of as functions of the Earth’s motion.

Now, think of those same observations as taken from another position on the Earth’s surface. A new set of curves will be generated that are a function the same motion of the Earth. But, the nature and position of those curves will be different than the curves traced by the observations from the first position.

These two sets of curves, give rise to a new function, that transforms the first set of curves into the second. That function reflects the effect of the curvature of the surface of the Earth. This function can not be visualized in the same way, as a set of curves, as in the case of the first two functions. This new type of function, a function of functions, is congruent with what Gauss and Riemann would refer to as a complex function.

In this example, a complex function is discovered that maps spherical functions into other spherical functions, which is another way of thinking about the concept of projection. The previous two discussions in this series, looked into types of complex functions that project spherical functions onto a surface of zero curvature (a plane), such as the gnomonic projection and the stereographic projection. These two complex functions transform the same curves from the sphere onto the plane, but in different ways.

The stereographic projection had the unique characteristic that the angles between great circle arcs on the sphere are not changed when projected onto the plane. This characteristic Gauss called conformal.

In his announcement to the first treatise on Hider Geodesy, Gauss points out that the curves conform in the infinitesimally small. However, in the large, the projection of the great circle arcs are magnified, the degree of magnification changes, depending on their position with respect to the point from which the projection is made. The experiment projecting circles tangent to the sphere, suggested in the last pedagogical, illustrated this point, at least intuitively.

In other words, if you think of the stereographic projection from Gauss’ standpoint, it is a special case of a complex function. A complex function that transforms curves on a sphere to curves on the plane, according to a law, that conforms in the infinitesimally small.

In the course of his geodesic investigations, Gauss was confronted with the requirement of discovering other complex functions that transformed functions on one surface to another. Rather than tackle each case separately, Gauss went into the matter more deeply, discovering the general principles on which these complex functions rested. This was the subject of his 1822 paper referred to in previous weeks, “General Solution of the Problem to so Represent the Parts of One Given Surface upon another Given Surface that the Representation shall be Similar, in its Smallest Parts, to the Surface Represented”. These investigations formed the foundation for Riemann’s theory of complex functions.

In his paper Gauss gives an example of such a problem from Higher Geodesy. In his geodetic survey, Gauss measured the area of a portion of the Earth’s surface, by laying out a series of triangles whose vertices were mutually visible. By measuring the angles between the lines of sight between these vertices, the area of the triangle could be computed. As this network of triangles was extended over the Kingdom of Hannover, the entire area of the entire region could be computed by adding up the areas of the smaller triangles in the network.

As discussed in previous weeks, the area of these triangles is a function of the shape of the surface on which they lie. If a spherical shape of the Earth is assumed, then the size of the triangle is a function of the sum of the angles comprising it multiplied by the diameter of the Earth.

Look back on our first example above. Between two positions on the surface of the Earth, a complex function characterizes the difference between the observed positions of the stars at those two positions. (For purposes of this example, consider the two positions as lying on the same meridian. Then the measurement of that complex function can be expressed as simply the difference in the angle of observation of the pole star between the two positions.) Based on an assumption about the size and shape of the Earth, the distance between the two positions along the surface of the Earth can be calculated.

The distance between those two positions can also be calculated by a geodetic triangulation carried out over the area of the Earth’s surface between the two positions. That distance, when compared with the enables us to test the original assumption of a spherical shape for the Earth. That type of measurement determined the shape of the Earth to be closer to an rather than a perfect sphere.

This confronted geodesist with the requirement of projecting those ellipsoidal triangles onto a sphere, conformally. Gauss was the first one to be able to solve this, by applying his general method of conformal projection. The method employed is analogous to Kepler’s measurement of planetary motion in an elliptical orbit, by the eccentric and mean anomalies, but with the use of complex functions, of the type described above.

In future weeks we will develop pedagogical exercises from Gauss’ examples, and then go on to a more thorough examination of Riemann’s revolutionary extension of Gauss’ discovery.

## Don’t Vote for Anyone Who Doesn’t Know Kepler

by Bruce Director

The foolishness of relying on pure mathematical models for the design and production of automobiles, nuclear weapons, or any other physical device, would be obvious to anyone with a minimal level of knowledge of the discoveries of Cusa, Kepler, Leibniz, Gauss, Riemann, et al. Unfortunately, such knowledge is virtually non-existent among the leaders of governments and businesses, today, as the frauds of the Mercedes A-class and the Cox report amply demonstrate. Fortunately, those who study the writings of Lyndon LaRouche need not suffer the afflictions of the aforementioned Lilliputians.

Take the case of Kepler’s discovery of the physical characteristics of planetary motion enunciated in his New Astronomy. As we demonstrate below, through their own words, Kepler demolished, nearly 400 years ago, the mathematical modelers of his day.

In the introduction of that work Kepler states:

“The reader should be aware that there are two schools of thought among astronomers, one distinguished by its chief, Ptolemy and the assent of the large majority of the ancients, and the other attributed to more recent proponents, although it is the most ancient. The former treats the individual planets separately and assigns cause to the motions of each in its own orb, while the latter relates the planets to one another, and deduces from a single common cause those characteristics which are found to be common to their motions. The latter school is again subdivided. Copernicus, with Aristarchus or remotest antiquity, ascribes to the translational motion of our home, the earth, the cause of the planets appearing stationary and retrograde. Tycho Brahe, on the other hand, ascribes this cause to the sun, in whose vicinity he says the eccentric circles of all five planets are connected as if by a kind of knot (not physical, of course, but only quantitative). Further, he says that this knot, as it were, revolves about the motionless earth, along with the solar body.

For each of these three opinions concerning the world there are several other peculiarities which themselves also serve to distinguish these schools, but these peculiarities can each be easily altered and amended in such a way that, so far as astronomy, or the celestial appearances, are concerned, THE THREE OPINIONS ARE FOR PRACTICAL PURPOSES EQUIVALENT TO A HAIR’S BREADTH, AND PRODUCE THE SAME RESULT.”

What Kepler is referring to is the fact that the observed motions of the stars, planets, sun, and moon, can be calculated equally by the three radically different mathematical models of Ptolemy, Copernicus, and Tycho Brahe.

The most elementary observations of the motions of heavenly bodies reveal two distinct motions. The so-called first motion, is the uniform daily movement across the sky of the sun, moon, stars, and planets from east to west. (Don’t take my word for it though. Go out an look for yourself!) The so-called second motion, is movement from west to east of the planets, sun, and moon, with respect to the fixed stars, over longer periods of time. Upon careful observation, this second motion is seen to be non-uniform. The planets, moon, and sun move slower and faster at different stages in the second motion, and, the planets, at times appear to stop and move backward with respect to the stars, at different stages in the course of the second motion.

The observation of these two motions is not the stuff of casual sense experience, but a characteristic of human reason. In the first chapter of the New Astronomy, Kepler says:

“The testimony of the ages confirms that the motions of the planets are orbicular. It is an immediate presumption of reason, reflected in experience, that their gyrations are perfect circles. For among figures it is circles, and among bodies the heavens, that are considered the most perfect. However, when experience is seen to teach something different to those who pay careful attention, namely, that the planets deviate from a simple circular pattern, it gives rise to a powerful sense of wonder, which at length drives men to look into causes.”

Neither Ptolemy, Copernicus, nor Tycho Brahe, however, ever laid claim to that “powerful sense of wonder,” of which Kepler speaks.

In the opening of the Almagast, Ptolemy says, “Those who have been true philosophers, Syrus, seem to me to have very wisely separated the theoretical part of philosophy from the practical…. For indeed Aristotle quite properly divides also the theoretical into three immediate genera; the physical, the mathematical, and the theological.”

Ptolemy goes on to say that man can know nothing certain of the theological nor physical:

“The theological because it is in no way phenomenal and attainable, but the physical because its matter is unstable and obscure, so that for this reason philosophers could never hope to agree on them; and meditating that only the mathematical, if approached enquiringly, would give its practitioners certain and trustworthy knowledge with demonstration both arithmetic and geometric resulting from indisputable procedures, we were led to cultivate most particularly as far as lay in our power this theoretical discipline.”

Having dispensed with any pretense that his theory had any physical reality, Ptolemy developed his now infamous system of intricate earth-centered cycles, eccentrics, and epicylces to mathematically calculate the positions of the planets, stars, moon, and sun, over time. While Ptolemy’s system can truthfully be called a fraud, the bigger frauds are those, who until this day, propounded this mathematical system, as physical hypothesis.

Copernicus replaced Ptolemy’s complicated system, with the simpler and more beautiful sun-centered system, where the earth and the planets move in perfect circles about a stationary sun. Nevertheless, this was a purely mathematical model. In the Introduction to his “On the Revolutions of the Heavenly Spheres,” Copernicus says:

“For it is the job of the astronomer to use painstaking and skilled observation in gathering together the history of the celestial movements, and then — since he cannot by any line of reasoning reach the true causes of these movements — to think up or construct whatever causes of hypotheses he pleases such that, by the assumption of these causes, those same movements can be calculated from the principles of geometry for the past and for the future. This artist is markedly outstanding in both of these respects; for it is not necessary that these hypotheses should be true, or even probable; but it is enough if they provide a calculus which fits the observations….”

As Kepler describes above, Tycho Brahe’s mathematical model had all the planets revolving around the sun, and this knot moving around a stationary Earth. But as Kepler says, Brahe’s system is not physical, but merely quantitative.

Since the systems of Ptolemy, Copernicus, and Brahe are all mathematically equivalent, and none lay claim to any physical reality, how can one distinguish which one is true? Only in the domain of physical measurement. This is precisely the revolutionary discovery that Kepler makes, following the path laid out by his mentor, Nicholas of Cusa.

Again, in the Introduction of the New Astronomy Kepler continues:

“My aim in the present work is chiefly to reform astronomical theory (especially of the motion of Mars) in all three forms of hypotheses, so that our computations from the tables correspond to the celestial phenomena. Hitherto, it has not been possible to do this with sufficient certainty. In fact, in August 1608, Mars was a little less than four degrees beyond the position given by calculation from the Prutenic tables. In August and September of 1593 this error was a little less than five degrees, while in my new calculation the error is entirely suppressed.

“… The eventual result of this consideration is the formulation of very clear arguments showing that only Copernicus’s opinion concerning the world (with a few small changes) is true, that the other two accounts are false, and so on.

“Indeed, all things are so interconnected, involved, and intertwined with one another that after trying many different approaches to the reform of astronomical calculations, some well trodden by the ancients and others constructed in emulation of them and by their example, none other could succeed than the one founded upon motions’ physical causes themselves, which I establish in this work.”

Readers of previous pedagogical discussions, and the Fidelio article on Gauss’ determination of the orbit of Ceres for will know something of Kepler’s discoveries. Isn’t it time we raised the level of thinking of the citizenry, so that they would demand such knowledge of their elected officials and designers of automobiles?

Newton’s World: No Love, Just Copulation

by Bruce Director

Several weeks ago we presented, in their own words, a demonstration that Kepler’s determination of the principles of planetary motion, demolished the Aristotelian methods of “mathematical modeling,” adhered to by Ptolemy, Brahe, and Copernicus. This week, we follow up with a further consequence of that demonstration: that all subsequent scientific inquiry that did not follow Kepler’s method was not just wrong, but fraudulent.

As presented in the previous discussion, Kepler, in the “New Astronomy,” set out to completely revolutionize astronomy (and all science) by putting it on a foundation of physical principles. As they testified themselves, Ptolemy, Copernicus, and Brahe were concerned only with developing formal descriptions of the observed motions of the planets. Truthfulness was limited to the logical-deductive consistency of those descriptions, and the consistency of those descriptions with observations. As Kepler stated, all three descriptions were equivalent “within a hair’s breadth,” but all three deviated from the observations by an amount greater than the margin of error associated with the capacity of the measuring instruments used for those observations.

The specific observed phenomena that concerned Kepler, Ptolemy, Brahe, and Copernicus, were the two unequal motions of the planets, observed by humankind since ancient times.

The first “inequality” was the observed non-uniform motion of the planets, in a cycle, from west to east, through the constellations of the zodiac. Each planet made this circuit in different lengths of time, and, as each travelled through its cycle, it appeared to move faster through certain constellations than others, that is, traversing a greater angular arc in the sky for a given time interval, depending on which constellation of the zodiac it was moving through.

The second “inequality” was the so-called “retrograde” motion, when the planet appeared to move from east to west through the zodiac. This was observed when the planet was rising in the east just as the sun set in the west. This configuration was known as “opposition.”

Ptolemy, Copernicus, and Brahe all described these phenomena with radically different geometrical constructions, but all held firm to the belief that these apparent non-uniform motions, were just that; “apparent,” not real. All three believed that the “true” motion of the planet had to be uniform circular motion. The two “inequalities” were simply optical illusions, owing to the complicated concoction of circles, that each had created.

Kepler took an entirely different approach:

“The testimony of the ages confirms that the motions of the planets are orbicular. It is an immediate presumption of reason, reflected in experience, that their gyrations are perfect circles. For among figures it is circles, and among bodies the heavens, that are considered the most perfect. However, when experience is seen to teach something different to those who pay careful attention, namely, that the planets deviate from simple circular path, it gives rise to a powerful sense of wonder, which at length drives men to look into causes.”

Driven by this “powerful sense of wonder,” Kepler looked into the causes. First he established the equality of the Ptolemaic, Brahean, and Copernican models. Then Kepler abandoned the false belief of embedded in all three models, that the “true” motion was uniform circular motion, and the non-uniform motion was simply apparent. Instead, Kepler took the apparent motion as the true. That is, that the planets actually did move non-uniformly. Once this conceptual bridge had been crossed, the geometrical construction of the planets moving on an orbit, about an eccentric and sweeping out equal areas in equal times, proceeded from the physical measurements themselves. The power that moved the planet, according to Kepler, had to be located at that eccentric.

Under this conception, the planet’s distance from the eccentric about which it was moving, varied continuously. That is, as the planet moved about it’s orbit, the distance from the planet to the eccentric was always getting longer or shorter, and consequently, the effect of the moving power was increasing as the distance decreased and diminishing as the distance increased. Then Kepler demonstrated that the moving power resided in the Sun, which was located at the eccentric point. When this conception was again tested against the physical measurements, Kepler refined his construction to an elliptical orbit with the Sun located at one of the foci. Later, Kepler demonstrated a third principle of planetary motion between the periodic times and the size of the orbit, mischaracterized today as his “Third Law.” (The reader can consult chapter’s 5-8 in the Summer 1998 Fidelio article on how Gauss Determined the Orbit of Ceres).

How absolutely banal, sterile, and fraudulent, is therefore, Newton’s resort to action at a distance according to the inverse square law. This is ass backwards. For Newton, the planetary motion is reduced to a copulation along the straight line connecting the planet to the Sun. The physical space time curvature of Kepler is eliminated. Only straight-line copulation remains.

So fraudulent is Newton’s view, that according to Riemann:

“Newton says: `That gravity should be innate, inherent, and essential to matter, so that one body can act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.’ See the third letter to Bently.”

Yet people continue to adhere to the false beliefs that underlie Ptolemy and Newton. With their asses facing the students, professors throughout the world present Newton’s straight-line copulation as the basis for planetary motion, despite the final burial of Newton by Gauss with his discovery of the orbit of Ceres. In his {Theoria Motus} Gauss says:

“The laws above stated differ from those discovered by our own Kepler in no other respect than this, that they are given in a form applicable to all kinds of conic sections … If we regard these laws as phenomena derived from innumerable and indubitable observations, geometry shows what action ought in consequence to be exerted upon bodies moving about the sun in order that these phenomena may be continually produced. In this way it is found that the action of the sun upon the bodies moving about it is exerted {as if} an attractive force, the intensity of which is reciprocally proportional to the square of the distance should urge the bodies towards the center of the sun. (emphasis supplied.)

Turn again to Kepler from the introduction of the {Mysterium Cosmographicum}:

“Though why is it necessary to reckon the value of divine things in cash like victuals? Or what use, I ask, is knowledge of the things of Nature to a hungry belly, what use is the whole of the rest of astronomy? Yet men of sense do not listen to the barbarism which clamors for these studies to be abandoned on that account. We accept painters, who delight our eyes, musicians, who delight our ears, though they bring no profit to our business. And the pleasure which is drawn from the work of each of these is considered not only civilized, but even honorable. Then how uncivilized, how foolish, to grudge the mind its own honorable pleasure, and not the eyes and ears. It is a denial of the nature of things to deny these recreations. For would that excellent Creator, who has introduced nothing into Nature without thoroughly foreseeing not only its necessity but its beauty and power to delight, have left only the mind of Man, the lord of all Nature made in his own image, without any delight? Rather, as we do not ask what hope of gain makes a little bird warble, since we know that it takes delight in singing because it is for that very singing that the bird was made, so there is no need to ask why the human mind undertakes such toil in seeking out these secrets of the heavens. For the reason why the mind was joined to the senses by our Maker is not only so that man should maintain himself, which many species of living things can do far more cleverly with the aid of even an irrational mind, but also so that from those things which we perceive with our eyes to exist we should strive towards the causes of their being and becoming, although we should get nothing else useful from them. And just as other animals, and the human body, are sustained by food and drink, so the very spirit of Man, which is something distinct from Man, is nourished, is increased, and in sa sense grows up on this diet for these things. Therefore as by the providence of nature nourishment is never lacking for living things, wo we can say with justice that the reason why there is such great variety in things and treasuries so well concealed in the fabric of the heavens, is so that fresh nourishment should never be lacking for the human mind and it should never disdain it as stale, nor be inactive, but should have in this universe an inexhaustible workshop in which to busy itself.”

Newton’s Gore

by Bruce Director

After reading the past two pedagogical discussions on this subject, there should be no doubt in your mind that Newton was a fraud. The question remains: why does Newton work? Not, why do Newton’s theories work — they don’t — but why does the fraud work?

The populist conspiracy theorist, or anyone else prone to superficial thinking might conclude that the fraud works through the suppression of Kepler. True, many of Kepler’s writings have been obscured over the ages, not widely published or translated, nor taught as original sources in secondary schools or universities. Nevertheless, they are available for any thinking person to obtain and study. Furthermore, the physical anomalies, from which the principles on which Kepler’s discoveries are based can be observed any night by anybody from any where on Earth.

No! it is not a lack of information, that keeps the fraud of Newton alive. Nor is the fraud perpetrated by controlling the purse strings of professors and scientists, or the raw political power of the British Royal Society, although that certainly is an element. None of that explains why generation after generation, Newton’s fraud is accepted willingly, to the point where victims of this fraud will hysterically defend it when challenged.

There is something more sinister involved, a vulnerability inside the mind of these wretched creatures that leads them to prefer the straight-line copulative world of Newton; to desire a world uncomplicated by the primacy of curvilinear action; and to yearn for a universe free of disturbing discontinuities.

To find this flaw, start with the report published in the May 31, 1999 briefing, quoting St. Augustine’s report from his Confessions of how his friend was drawn, against his better judgement, into lusting for the savagery of the Roman Circus. This begins to approximate the mindset that draws the unsuspecting dupe into Newton’s world.

Or, turn to the insightful allegory “Mellonta Tauta” of Edgar Allen Poe, whose protagonist reports:

“Do you know that it is not more than a thousand years ago, since the metaphysicians consented to relieve the people of the singular fancy that there existed but {two possible roads for the attainment of Truth!} Believe it if you can! It appears that long, long ago, in the night of Time, there lived a Turkish philosopher (or Hindoo possibly) called Aries Tottle. This person introduced, or at all events propagated what was termed the deductive or {a priori} mode of investigation. He started with what he maintained to be axioms or `self-evident truths,’ and thence proceeded `logically’ to results. His greatest disciples were one Neuclid and one Cant. Well, Aries Tottle flourished supreme until the advent of one Hog, surnamed the `Ettrick Shepherd,’ who preached an entirely different system, which he called the {a posteriori} or {inductive}. His plan referred altogether to Sensation. He proceeded by observing, analyzing and classifying facts — {instantiae naturae}, as they were affectedly called — into general laws. Aries Tottle’s mode, in a word, was based on {noumena}; Hog’s on {phenomena}. Well, so great was the admiration excited by this latter system that, at its first introduction, Aries Tottle fell into disrepute; but finally he recovered ground, and was permitted to divide the realm of Truth with his more modern rival. The savants now maintained that the Aristotelian and Baconian roads were the sole possible avenues to knowledge. `Baconian,’ you must know, was an adjective invented as equivalent to Hog-ian and more euphonious and dignified.

“Now, my dear friend, I do assure you, most positively, that I represent this matter fairly, on the soundest authority; and you can easily understand how a notion so absurd on its very face must have operated to retard the progress of all true knowledge — which makes its advances almost invariably by intuitive bounds. The ancient idea confined investigation to {crawling} and for hundreds of years so great was the infatuation about Hog especially, that a virtual end was put to all thinking properly so called. No man dared utter a truth for which he felt himself indebted to his Soul alone. It mattered not whether the truth was even {demonstrably} a truth, for the bullet-headed {savants} of the time regarded only {the road} by which he had attained it. They would not even look at the end. `Let us see the means,’ they cried, `the means!’ If, upon investigation of the means, it was found to come neither under the category Aries (that is to say Ram) or under the category Hog, why then the {savants} went no farther, but pronounced the `theorist’ a fool, and would have nothing to do with him or his truth….

“Now I do not complain of these ancients so much because their logic is, by their own showing, utterly baseless, worthless and fantastic altogether, as because of their pompous and imbecile proscription of all {other} roads of Truth, of all {other} means for its attainment than the two preposterous paths — the one of creeping and the one of crawling — to {which} they have dared to confine the Soul that loves nothing so well as to {soar}.

“By the by, my dear friend, do you not think it would have puzzled these ancient dogmaticians to have determined by {which} of their two roads it was that the most important and most sublime of {all} their truths was, in effect, attained? I mean the truth of Gravitation. Newton owed it to Kepler. Kepler admitted that his three laws were {guessed at} — these three laws of all laws which led the great Inglitch mathematician to his principle, the basis of all physical principle — to go behind which we must enter the Kingdom of Metaphysics. Kepler guessed — that is to say, {imagined}. He was essentially a `theorist’ — that word now of so much sanctity, formerly an epithet of contempt. Would it not have puzzled these old moles, too, to have explained by which of the two `roads’ a cryptographist unriddles a cryptograph of more than usual secrecy, or by which of the two roads Champollion directed mankind to those enduring and almost innumerable truths which resulted from his deciphering the Hieroglyphics?”

For the moment, no more need be said.

## Hypergeometric Curvature

by Bruce Director

Let us turn our investigations to the domain of manifolds of a Gauss-Riemann hypergeometrical form. There is no need, as too often happens, for your mind to glaze over as you read the above mentioned words. Lyn has given us ample guidance for this effort, most recently in his memo on non-linear organizing methods.

Over the next few weeks, let us set a course, by way of several preliminary exercises that will shift our investigations from manifolds of constant curvature, that we’ve been looking at for the last couple of months, to investigations of manifolds of non-constant curvature.

A WARNING: these exercises should not be taken as some type of definition of the concepts involved, any more than bel canto vocalization should be taken as a substitute for singing classical compositions. However, without the former, the latter is unattainable.

As a first start, conduct the following experiment, that was alluded to in the previous pedagogical discussion on the pentagramma mirificum:

Think of a surface of zero curvature, represented as a flat piece of paper. This manifold is characterized by the assumption of infinite extension in two directions. The intersection of these two infinitely extended directions produces a singularity: a right angle, to which all geodetic action is referred.

Now, draw a right triangle, labelling the vertices BAC, with the right angle at A. Extend the hypothenuse BC to some arbitrary point D. (BCD will all lie on the same line.) At D, draw a line perpendicular to line BCD, and extend line AC until it intersects the perpendicular from D. Label that point of intersection E. (You will now have produced two right triangles, with a common vertex at C. The extension of leg AC of the first triangle, will form the hypothenuse CE of the second triangle CDE. Continue this action by extending line ACE to some point F. At F, produce a perpendicular line, and extend leg DE of triangle CDE until it meets the new perpendicular at some point G.

Now you will have three right triangles, BAC, CDE, and EFG forming a kind of chain. Continue to produce this chain of right triangles, by extending the hypothenuse EF of triangle EFG to some arbitrary point H. Draw a perpendicular to H and extend leg FG until the two meet at some point I. Now the chain has four triangles in it.

Keep adding to the chain of triangles in the same manner. You will notice that after every three triangles, the chain “turns” a corner. After the chain has eight triangles, if the appropriate lengths were chosen, the triangle will close. The closed chain of triangles, will resemble two intersecting rectangles. (We leave it the reader to discover what the appropriate lengths are for the chain to exactly close. As you will discover, the fundamental point is not lost, even if arbitrary lengths are used. In that case, the orientation of the 9th triagle will be identical to the 1st.)

Now produce the same action on a sphere, i.e. a surface of constant positive curvature. Begin with a right spherical triangle BAC. Extend its hypothenuse to some point D. At D, draw an orthogonal great circle arc. Extend the side AC until it intersects the orthogonal arc you just drew from D. Continue producing this chain of spherical triangles. You will discover, that the chain of right triangles on the sphere, closes after five “links” have been produced. In other words, the pentagramma mirificum!

(If each hypothenuse is extended to an arc length of 90 degrees, the chain will perfectly close after 5 links. If an arbitrary arc length is used, as in the plane, the chain will not perfectly close, but the orientation of the 6th and 1st triangle will be the same. On a sphere, the lengths need not be arbitrary, as a 90 degree arc length is determined by the characteristic curvature of the sphere. On a plane, no such ability to determine length exists.

Now, think about the results of this experiment. The same action was performed on a manifold of zero-constant curvature and a manifold of constant positive curvature. The same action on, two different manifolds, produces two distinctly different periodicities. What in the naive imagination’s conception of the plane and sphere, accounts for two completely different periodicities arising from exactly the same process?

Now try a second experiment:

Stand in a room fairly close to two walls. Mark a dot on the ceiling directly above your head. Point to that dot and rotate your arm down 90 degrees so that you’re pointing to a place on the wall directly in front of you. Mark a dot on that wall. Point to that dot, and rotate your arm 90 degrees horizontally to a point on the wall directly to your right (or left). Mark a dot on that wall at that point.

As presented in previous pedagogicals, the manifold of action, that generated the positions of these three dots, is characteristic of a surface of constant positive curvature, i.e. a sphere. The three dots are vertices of a spherical equilateral triangle.

Now, take some string and masking tape and connect the dots to one another with the string. Since the strings form the shapes of catenaries, those same dots are now the vertices of a negatively curved triangle.

Finally, in your mind, connect the dots with straight lines, and those same dots represent vertices of a Euclidean triangle.

From this construction, the same three positions lie on three different surfaces.

But, there is also another type of “surface” represented in this experiment. A hypergeometric manifold characterized by the change in curvature from negative, to zero, to positive curvature.

This is not simply a trivial class room experiment. In our previous discussions, we generated the concept of a sphere, as a manifold of measurement of astronomical observations. Instead of being in a room, the three dots can be thought of as stars, whose positions on the celestial sphere are 90 degrees apart.

But, couldn’t the relationship of these three stars, also be conceived to lie on a surface of constant negative curvature? In 1819, Gauss’ collaborator Gerling forwarded to Gauss the work of a friend of his named Schweikart, a professor of law whose avocation was mathematics and astronomy. Schweikart had developed a conception, that he called, “Astralgeometrie”, that conceived of the spatial relationship among astronomical phenomena as a negatively curved manifold. Gauss replied, that Schweikart’s ideas gave him, “uncommonly great pleasure” to read and agreed with almost all of it. In his reply, Gauss added a few additional ideas to Schweikart’s hypothesis.

It should come as no surprise, that Gauss would receive Shweikart’s work so warmly. Three years earlier, Gauss had expressed an even more advanced notion, in his April 1816 letter Gerling, that we have cited several times before, most recently two weeks ago:

“It is easy to prove, that if Euclid’s geometry is not true, there are no similar figures. The angles of an equal-sided triangle, vary according to the magnitude of the sides, which I do not at all find absurd. It is thus, that angles are a function of the sides and the sides are functions of the angles, and at the same time, a constant line occurs naturally in such a function. It appears something of a paradox, that a constant line could possibly exist, so to speak, a priori; but, I find in it nothing contradictory. It were even desirable, that Euclid’s Geometry were not true, because then we would have, a priori, a universal measurement, for example, one could use for a unit of space (Raumeinheit), the side of an equilateral triangle, whose angle is 59 degrees, 59 minutes, 59.99999… seconds.”

I’m sure you found Gauss’ choice of a triangle whose angle is 59 degrees, 59 minutes, 59.99999… seconds curious. But, think about it in the context of the above reference to a hypergeometric manifold characterized by a change from negative to zero, to positive curvature. The surface of zero curvature, is nothing more than a singularity, in that hypergeometric manifold. The sum of the angles of a triangle in a manifold of negative curvature will be less than 180 degrees. The 60 degree equilateral triangle is the maximum. On a surface of positive curvature, the sum of the angles of a triangle is always greater than 180 degrees. The 60 degree equilateral triangle in this manifold, is the absolute minimum.

The triangle Gauss proposes for an absolute length, does not exist in a manifold of negative curvature, nor in a manifold of positive curvature. And, on a surface of zero curvature, it can no longer define an absolute length. On the other hand, in a hypergeometric manifold, that characterizes the change from negative, to zero, to positive curvature, such a triangle represents, a unique singularity, a maximum and a minimum, existing in the infinitessimally small interval, in between two mutually distinct curvatures.

Enjoy the exercises. We’ll be back next week.

The Case For Knowing It All

by Bruce Director

A common mistake can occur, when replicating Gauss’ method for determining the Keplerian orbit of a heavenly body from a small number of observations within a small interval of the orbit, that has wider general implications. The error often takes the form, of asking the rhetorical question, “What did Gauss do, exactly?” and, answering that question, with a rhetorical step-by-step summary of a procedure for calculating the desired orbit. In fact, Gauss himself never published, or wrote down any such procedure. Gauss determined the orbit of Ceres in the summer of 1801, and communicated only the result of that determination, so that astronomers watching the sky could re-discover the previously observed asteroid. It wasn’t until 8 years later that Gauss, after repeated requests, published his “Summary Overview,” and a year after that, his “Theory of the Motion of the Heavenly Bodies Moving About the Sun In Conic Sections.”

Both these works, refrain completely from presenting any step-by-step procedure — because no such procedure existed. Instead, Gauss presented, first in summary form, than in a more expansive way, the totality of interconnected principles that underlay the motion of bodies in the solar system. These principles are not a collection of independent functions that are mutually interdependent. Rather, that mutual connectedness is itself a function, a representation of a higher principle that governs planetary motion.

To illustrate this point, think of Kepler’s principles of planetary motion, maliciously mis-characterized as Kepler’s three laws. The elliptical nature of the orbit, the constant of proportionality for each orbit (the “equal area” principle), and the constant of proportionality between the periodic times and the semi-major axis of the elliptical orbits, were each demonstrated by Kepler as a valid principle governing planetary motion. But (as those who’ve worked through the Fidelio article will recognize), all three principles are inseparably linked in each small interval of every planetary orbit. It is the functional relationship among these principles, the “hypergeomtric” relationship, that is the essence of Kepler’s discovery.

It is the “disassembly” of this hypergeometric relationship, into separate independent functions, that has been the hysterical obsession of the oligarchy and its lackeys, from Newton, to Euler, to today’s academics.

Leibniz, in a letter to Huygens exposed this hoax from the get go:

“For although Newton is satisfactory when one considers only a single planet or satellite, nevertheless, he cannot account for why all the planets of the same system move over approximately the same path, and why they move in the same direction….”

Or, from another angle: Nearly 20 years after his discovery of the orbit of Ceres, Gauss took on the task of measuring the Kingdom of Hannover, by means of a geodetic triangulation. In the course of this investigation, which had many practical implications, Gauss demonstrated a similar “hypergeometric” relationship. Each triangle he measured was “infinitesimally” small with respect to the entire Earth’s surface, and the deviation of those triangles from flat ones was also small. As the network of triangles was extended, however, the small deviation in each individual triangle, became an increasingly significant factor in the measurement of the larger area covered by the connected network of these triangles. Not only did the area measured deviate from flat, but it also deviated from a spherical surface, and more closely resembled an ellipsoidal surface. Furthermore, Gauss discovered an “infinitesimally” small deviation from the astronomical observation of his position on the Earth’s surface, and the position determined by his triangulation. This led Gauss to the discovery of the deviation of the Earth’s surface, from one of regular non-constant curvature, such as an ellipsoid, to a surface of irregular, non- constant curvature, that today is called the Geoid.

This defines a functional relationship of the measurement of the relatively “infinitesimally” small triangles, and the multiple surfaces on which these measurements were performed. That is, each triangle measured, had to be thought of simultaneously as being on a surface of zero-curvature (flat), constant curvature (spherical), regular non-constant curvature (ellipsoidal), and irregular non-constant curvature (the Geoid). The characteristics of each triangle changes from surface to surface. But, in the real world, these surfaces are not independent surfaces, simply overlaid on top of each other. There is a functional relationship among them. Gauss’ genius was to recognize, not only the interaction between the characteristic of curvature of the surface, and the characteristic of the triangles measured, but also the functional relationship that transformed one surface into another.

Or, from an even different angle: In 1832, after nearing the completion of his geodetic survey, Gauss published the results of the work he had been doing along the way. In his second treatise on bi-quadratic residues, Gauss extended the concept of prime numbers into the complex domain, transforming Eratosthenes’ Sieve. Gauss showed that the characteristics of prime numbers, were also a function of the nature of the surface, such that, for example, 5 is transformed from a prime to a composite number. The number 5 exists in both domains, but it’s nature changes, as the domain changes. The number 5 is not two separate independent numbers. Again there is a functional relationship between these two domains, the transformation, that provokes our minds to a higher mode of cognition.

The above three examples, presented in summary form, have been elaborated in previous pedagogical discussions, and will be further elaborated in future ones. The intent in presenting this summary juxtaposition, is to provoke some thought on the functional relationship among these three. They are not three independent concepts. There is a connection, whose active contemplation, gives rise to a conception of functional relationship, that governs the generation of each concept.

As Lyndon LaRouche has wisely advised us, “If you want to know anything, you have to know everything.”

## Higher Arithmetic as a Machine Tool

by Bruce Director

Last week’s pedagogical discussion ended with the provocative question: “If there exists no grand mathematical system which can combine and account for the various cycles, then how can we conceptualize the `One’ which subsumes the successive emergence of new astronomical cycles as apparent new degrees of freedom of action in our Universe? How do we master the paradoxical principle of Heraclitus, that `nothing is constant except change?'”

This problem was attacked in a very simple and beautiful way by C.F. Gauss, using purely the principles of higher arithmetic, in his determination of the Easter date. Since the last conference presentation, I have received several requests, to elaborate more completely the derivation of Gauss’ algorithm. While the development of Gauss’ program requires no special mathematical skills other than simple arithmetic, it does require the conceptual skills of higher arithmetic, i.e., the ability for the mind to unify an increasingly complex Many into a One. This is a subjective question. We are not looking for one mathematical formula, but a series of actions, which, when undertaken, enable our minds to wrestle a seemingly unwieldy collection of incommensurable cycles into our conceptual grasp. In a certain sense, we are designing and building a machine tool to do the job, but only the entire machine can accomplish the task. No single part, or collection of parts, will be sufficient. The whole machine includes not only the “moving parts,” but the concepts behind those moving parts. All this, the parts and the concepts, must be thought of as a “One,” or else, the machine, i.e., your own mind, comes to screeching halt, while the earth, the moon, the sun, and the stars, continue their motion, in complete defiance of your blocking.

Over the next few weeks, we will re-discover Gauss’ construction. But, in order to build this machine, you must be willing to get your hands dirty and break a sweat, make careful designs, cut the parts to precision, lift heavy components into place, and finally apply the energy (agape) necessary to get the machine moving and keep it moving.

In the beginning of his essay, “Calculation of Easter,” published in the August 1800 edition of Freiherrn v. Zach’s “Monthly Correspondence for the Promotion of News of the Earth and Heavens,” Gauss states:

“The purpose of this essay, is not to discuss the usual procedure to determine the Easter date, that one finds in every course on mathematical chronology, and as such, is easy enough to satisfy, if one knows the meaning and use of the customary terms of art, such as Golden Number, Epact, Easter Moon, Solar Cycle, and Sunday Letter, and has the necessary helping tables; but this task is to give, independently from those helping conceptions, a purely analytical solution based on merely the simplest calculation-operations. I hope, this will not be disagreeable, not only to the mere enthusiast who is not familiar with those methods, or for the case where one wishes to determine the Easter date, under conditions in which the necessary helping devices are not at hand, or for a year which cannot be looked up in a calendar; but it also recommends itself to the expert by its simplicity and flexibility.”

This article was published after Gauss had completed, and was awaiting publication of the “Disquistiones Arithmeticae.” Of the principles we will develop here, Gauss says:

“The analysis, by means of which the above formulas are founded, is based properly on the foundations of {Higher Arithmetic}, in consideration of which I can refer presently to nothing written, and for that reason it cannot be freely presented here in its complete simplicity: in the mean time, the following will be sufficient, in order to lay the foundation of the direction of the concept and to convince you of its correctness.”

Gauss’ choice of the problem of determining the Easter date, to demonstrate the validity of the principles of his Higher Arithmetic, is not without a healthy amount of irony, but the resulting calculation was by no means Gauss’ only goal. As with LaRouche’s current program of pedagogical exercises, Gauss recognized the effectiveness for increasing the conceptual powers of the human mind, of working through specific examples, which demonstrate matters of principle. Gauss continued this approach in all his work, demonstrating new principles as he conquered one problem after another. Gauss repeatedly found that in these matters of principle, connections were discovered between areas of knowledge which were previously thought to be unrelated.

From the earliest cultures, the various cycles described last week were accounted for separately, and their juxtaposition was studied with aid of the different tables and calculations Gauss mentioned above. These methods were adequate for determining the date of Easter from year to year. Gauss’ calculation is purely a demonstration of the power of the human mind, to create a new mathematics, capable of bringing into a “One” that which the previous state of knowledge considered unintelligible. For that reason, it suits our present purpose.

To begin, we should think about the problem we intend to work through: To determine the date of Easter for any year. Easter occurs on the first Sunday, after the first full Moon (called the Paschal Moon) after the Vernal Equinox. This entails three incommensurable astronomical cycles: the day, the solar year, and the lunar month; and one socially-determined cycle, the seven-day week.

Now look more closely at what this “machine-tool” must do:

1. It must determine the number of days after the vernal equinox, on which the Paschal Moon occurs. This changes from year to year. So the machine must have a function, which modulates the solar year (365.24 days) with the lunar month (29.53 days).

2. Once this is determined the machine must also determine the number of days, remaining until the next Sunday.

The incommensurability of the solar year and the lunar month is an ancient conceptual problem, upon whose resolution man’s potential for economic progress rested. If one relied solely on the easier-to-see lunar month, the seasons (which result from changes of the position of the earth with respect to the sun) will occur at different times of the year, from one year to the next. On the other hand, if one relies on the solar year, some intermediate division between the day and year is necessary, to measure smaller intervals of time. Efforts to combine both the lunar cycle, and the solar cycle, linearly into one calendar, creates a complicated mess. The Babylonian-influenced Hebrew calendar is an example, requiring a special priestly knowledge just to read the calendar. Shortly after the publication of the Easter formula, Gauss applied the same method to a much more complex chronological problem, the determination of the first day of Passover, and in so doing, subjugating the Babylonian lunisolar calendar to the powers of Higher Arithmetic.

In 423 B.C., the Greek astronomer Meton reportedly discovered that 19 solar years contained 235 lunar months. This is the smallest number of solar years, that contain an integral number of lunar months. There is evidence that other cultures, including the Chinese, discovered this same congruence earlier. By the following simple calculation, we can re-discover Meton’s discovery. One solar year is 365.2425 days. 12 lunar months is 354.36 days, (12 x 29.53) or 11 days less than the solar year. This means that each phase of the moon will occur 11 days earlier than the year before, when compared to the solar calendar.

(For example, if the new moon falls on January 1, then after 12 lunar months, a new moon will fall on December 20 — 11 days before the next January 1. The next new moon will occur on January 19, 19 days after the next January 1.)

One solar year contains 12.368 lunar months (365.2425 / 29.530). In 19 years, there are 6939.6075 days (365.2425 x 19). In 19 years of 12.368 lunar months, there are 6939.3137 days (19 x 12.368 x 29.530). That is, if you take a cycle of 6939 days, or 19 solar years, the phases of the moon and the days of the solar year become congruent.

Despite Meton’s discovery, the Greek calendar was still encumbered by a failed effort to combine the lunar months and solar year into a single linear calendar cycle. Since 12 lunar months, are 11 days short of the solar year, the Metonic calendar, like the Babylonian influenced Hebrew calendar, required the intercalation (insertion) of leap months in years 3, 5, 8, 11, 13, and 16 of the 19-year cycle.

In his “History,” Herodotus remarks on the inferiority of the Greek method over the Egyptians, whose calendar was based only on the harder-to-measure solar year. “But as to human affairs, this was the account in which they all agreed: the Egyptians, they said, were the first men who reckoned by years and made the year consist of twelve divisions of the seasons. They discovered this from the stars (so they said). And their reckoning is, to my mind, a juster one than that of the Greeks; for the Greeks add an intercalary month every other year, so that the seasons agree; but the Egyptians, reckoning thirty days to each of the twelve months, add five days in every year over and above the total, and thus the completed circle of seasons is made to agree with the calendar.”

The oligarchical view of this matter is expressed by the Chorus-Leader in Aristophanes, “The Clouds”:

“As we prepared to set off on our journey here,
The Moon by chance ran into us and said she wanted
To say hello to all the Athenians and their allies,
but she’s most annoyed at your treating her so shamefully
despite her many evident and actual benefactions.
First off, she saves you at least ten drachmas a month in torches:
that’s why you all can say, when you go out in the evening,
No need to buy a torch, my boy, the moonlight’s fine!
She says she helps in other ways too. But you don’t keep
your calendar correct; it’s totally out of sync.
As a result, the gods are always getting mad at her,
whenever they miss a dinner and hungrily go home
because you’re celebrating their festival on the wrong day,
or hearing legal cases or torturing slaves instead of sacrificing.
And often, when we gods are mounring Memnon or Sarpedon,
you’re pouring wine and laughing. That’s why Hyperbolus,
this year’s sacred ambassador, had his wreath of office
blown off his head by us gods, so that he’ll remember well
that the days of your lives should be reckoned by the Moon.”

In 46 B.C., with the adoption of the Julian calendar, all attempts to incorporate the lunar cycle into the calendar were abandoned. But, it wasn’t until Gauss’ development of higher arithmetic, ironically based on a re-working and non-linear extension of classical Greek astronomy and geometry, that man had the ability to encompass the seemingly incommensurable lunar month and solar year into a One.

With these discoveries in mind, we can begin to construct the first components of the machine, which will determine the number of days from the vernal equinox, to the Paschal Moon. If we fix the vernal equinox at March 21, our first component must determine some number D, which, when added to March 21, will be the date of the Paschal Moon. (March 21 was the date set at the Council of Nicea. The actual Vernal Equinox, can sometimes occur in the late hours of March 20, or the early hours of March 22.) The Paschal Moon will occur on one of 30 days, the earliest being March 21, the latest being April 19. The variation from year to year, among these 30 possible days, is a reflection of the 19- year Metonic cycle. So, our machine, must make two cycles, the 19-year Metonic cycle, and this 30-day cycle into a One.

This requires some thinking. Since 12 lunar months are 11 days less than the solar year, any particular full moon will occur 11 days earlier than the year before. Naive imagination tells us that if we set our machine on any given year, all it need do is subtract 11 days to find the Paschal Moon on the next year. But we have a boundary condition to contend with. The Paschal Moon can never occur before March 21. So, when the Paschal Moon occurs in March, and our machine subtracts 11 days, to get the date of the Paschal Moon the following year, the new date will be before March 21. That will do us no good at all.

To determine the date of the Paschal Moon from one year to the next, our machine must do something different when the Paschal Moon occurs in March, than when it occurs in April. When the Paschal Moon occurs in April, the machine must subtract 11 days, to determine the date for the following year. But when it occurs in March, the machine must add 19 days to determine the date for the following year.

To construct this component of the algorithm, Gauss began with a known date, and abstracted the year-to-year changes, with respect to that date. In reference to the 19-year Metonic cycle, he chose to begin the calculation with the date of the Paschal Moon in the first year of that cycle (i.e., those years which, when divided by 19, leave 0 as a remainder, or are congruent to 0 relative to modulus 19). In the 18th and 19th centuries, that date was April 13, or March 21 + 23 days.

For clarity, we can make the following chart:

Year Residue Paschal Moon # Days Aft. Equinox (D)

(Mod 19)

1710 0 April 13 23 days 1711 1 April 2 23 – 11 days 1712 2 March 22 23 – (2 x 11) 1713 3 April 10 23 – (2 x 11) + 19 1714 4 March 30 23 – (3 x 11) + 19

(The reader is encouraged to complete this entire chart. When you do this notice the interplay between the 19 year, and 30 day cycles.)

From the chart, you should be able to see the relevant oscillation. For example, for year 1713, were we to have subtracted another 11 days from the year before, we’d arrive at the date of March 11. A full moon certainly occurred on that day, but it wasn’t the Paschal Moon, because March 11 is before the Vernal Equinox. The Paschal Moon, in the year 1713, occurred 30 days later than March 11, on April 10. (March 22 – 11 + 30; or March 22 + 19)

The number of days added or subtracted changes from year to year, in a seemingly non-regular way. What is constant is change. But this step-by-step process, is really no different than if we had a series of tables.

Gauss’ next step, is to transform the two actions, subtracting 11 days or adding 19 days, into one action. There are many ways this can be done. The determination of the appropriate one, is a matter of analysis situs, and involves one of the most important methods of scientific inquiry: {inversion}. The principle of inversion is characteristic of all Gauss’ work. It is one thing to be given a function, and then calculate the result. The inverse question is much more difficult. Given a result, what are the conditions which brought about that result? In the latter case, there are many possible such conditions, which cannot be ordered without consideration of higher dimensionalities. (This subject will be treated more in future pedagogical discussions.)

Our immediate problem can be solved, if we think about it from the standpoint of inversion. All the year-to-year differences between the dates of the Paschal Moon, are either congruent relative to modulus 11 or modulus 19. But neither of these moduli are relevant for the task at hand. A different modulus must be discovered, which is not self-evident from the chart, but is evident from the higher dimensionality of the complete process. As discovered earlier, the Paschal Moon occurs on one of 30 days between March 21, and April 19. We need to discover a means, under which the oscillation of the dates of the Paschal Moon, can be ordered with respect to modulus 30. If we number these days 0-29, the numbers 0 to 29 each represent different days, and are all non-congruent relative to modulus 30.

Gauss chose to combine the two actions into one, by adding 19 days to {every} year, and subtracting 30 days from those years in which the Paschal Moon occurs in April. (For example, in our chart above, the year 1711 would be calculated: 23 + 19 – 30; the year 1712 would be calculated, 23 + (2 x 19) – (2 x 30).

Since all numbers whose differences are divisible by 30, are all congruent relative to modulus 30, adding or subtracting 30 days from any interval, will not change the result. Gauss has transformed this problem into a congruence relative to a single modulus: 30. So the first component of our machine takes the year, finds the residue, multiplies that by 19, adds 23, divides by 30 and the remainder is the number of days from the Vernal Equinox to the Paschal Moon.

Or in Gauss’ more condensed language: Divide the year by 19 and call the remainder a. Then divide (23 + 19a) by 30 and call the remainder D. Add D to March 21 to get the date of the Paschal Moon.

No mountain was ever climbed that didn’t require some sweat. Or, put another way, in order to build the Landbridge, you have to move some dirt.

Next week: From the Paschal Moon to Easter.

Higher Arithmetic as a Machine Tool–Part II

by Bruce Director

Last week we completed the first step of the development of Gauss’ algorithm for calculating the Easter date, using the principles of Higher Arithmetic. This week we continue the climb. Those experienced in climbing mountains are aware, that as one approaches the peak, the climb often steepens, requiring the climber to find a second burst of energy. Even though last week’s climb might have required some exertion, you’ve had a week’s rest, and a national conference in the intervening period. Armed with the higher conceptions of man expressed by Lyn and Helga at the conference, everyone is well-equipped to complete this climb.

Again it is important to keep in mind, that the determination of the date of Easter was not a goal in itself for Gauss. Rather, Gauss understood that working through problems, which required the discovery of new principles, was the only way to advance human knowledge.

Last week, we worked through the first part of the task of determining the date of Easter. Since Easter is the first Sunday after the first full moon, after the vernal equinox, the first job of our machine tool, is to determine the date of the first full moon. This requires bringing into a One, three astronomical cycles: the day, the lunar month, and the solar year. The second part of the job, to determine the number of days from the Paschal Moon until the next Sunday, requires bringing into a One, various imperfect states of human knowledge.

It was a major step forward, for society to abandon all attempts to reconcile the lunar and solar years into one linear calendar, and adopt the solar year, as the primary cycle on which the calendar was based. The conceptual leap involved was to base the calendar on the more difficult to determine solar year, instead of the easier to see lunar months. The implications of this conceptual leap for physical economy are obvious. What is worth emphasizing here, is, that this is a purely subjective matter, whose resolution determines physical processes. This development, however, was not without its own problems.

While the disaster of trying to reconcile the lunar and solar cycles, becomes evident within the span of several years, the problems of the solar calendar, don’t become significant within in the span of a single human life.

As discussed last week, the solar year is approximately 365.24 days. In 46 B.C., the calendar reform under Roman Emperor Julius Caesar, set the solar year at 365.25 days, which was reflected in the calendar, by three years of 365 days, followed by a leap year of 366 days. The number of days in this arrangement, would coincide every four years. Under this arrangement, man has imposed on the astronomical cycles, a new four-year cycle. From the standpoint of Gauss’ Higher Arithmetic, leap years are congruent, in succession to 0 relative to modulus 4, followed by non-leap years congruent to 1, 2, or 3 relative to modulus 4.

Like all oligarchs who delude themselves that their rule will last forever, Julius Caesar’s arrogance of ignoring the approximately .01 discrepancy between his year, and the actual astronomical cycle, became evident long after his Empire had been destroyed. This .01 discrepancy, while infinitesimal with respect to a single human life, becomes significant with respect to generations, causing the year to fall one day behind every 187 Julian years. By the mid-16th century, this discrepancy had grown to 11 days, so the astronomical event known as the vernal equinox was occurring on March 10th instead of March 21st. The economic implications of such a discrepancy is obvious.

This lead to the calendar reforms of Pope Gregory XIII in 1587. In the Gregorian calendar, the leap year is dropped every century year, except those century years divisible by 400. This decreases the discrepancy of the .01 day, but doesn’t eliminate it altogether. In order to get the years back into synch with the seasons, Pope Gregory dropped 11 days from the year 1587. Other countries reformed their calendar much later, having to drop more days, the longer they waited. The Protestant states of Germany, where Gauss lived, didn’t adopt the calendar reform until the early 1700s. The English didn’t change their calendar until 1752. The Russians waited until the Bolshevik revolution.

The other human cycle involved in this next step of the problem is the seven-day week. There is no astronomical cycle which corresponds to the seven-day week. While the Old Testament’s Exodus, attributes the seven-day week to God’s creation of the universe, Philo of Alexandria, in his commentaries on the Creation, cautions that this cannot be taken literally. Philo says the Creation story in Genesis 1, must be thought of as an ordering principle, not a literal time-table. Here is another example of what Lyn has discussed about the unreliability of a literal reading of the Old Testament. The idea that creation took seven days, shows up in Exodus, contradicting the conception of an ordering principle of Creation in Genesis 1.

Of importance for our present problem, is that, the seven-day weekly cycle runs continuously, and independently, from the cycles of the months, (either calendar or lunar) and the years. What emerges is a new cycle which has to be accounted for. Each year, the days of the week occur on different dates. For example, if today is Saturday, September 6, next year, September 6 will be on a Sunday. However, when a leap year intervenes, the calendar dates move up two days. This interplay between the seven-day week and the leap year, creates a 28-year cycle, before the days of the week and the calendar dates coincide again. This cycle also has to be accounted for in Gauss’ algorithm.

So, to climb that last step, from the Paschal Moon to Easter, we have to bring into a One, these two human cycles, the leap year, and the seven-day week.

Before going any further, one must first remember a principle of Higher Arithmetic. Under Gauss’ conception of congruence, it is the {interval} between the numbers, on which the congruence is based, not the numbers themselves. We are relating numbers by their intervals. Consequently, when we add or subtract multiples of the modulus to any given number, the congruence relative to that modulus doesn’t change. For example, 15 is congruent to 1,926 relative to modulus 7. The interval between 15 and 1,926 (1,911) is divisible by 7. If, for example, we subtract 371 (7×53) from 1,926, the result will still be congruent to 15. The reader should do several experiments with this concept, in preparation for what follows.

It were useful to restate here Gauss’ entire algorithm:

Divide the year by 19 and call the remainder a

Divide the year by 4 and call the remainder b

Divide the year by 7 and call the remainder c

Divide 19a+23 by 30 and call the remainder d

(This was discovered last week)

Divide 2b+4c+6d+3 by 7 and call the remainder e

(This is today’s task.)

The number of days from the Paschal Moon until Easter Sunday can be at least 1 and at most 7 days. Because Easter is the first Sunday {after} the first full moon, which follows the Vernal Equinox, the earliest possible date for Easter is March 22. Therefore, Easter will fall on March 22 + d (the number of days to the Paschal Moon) + E (the number of days until Sunday.) E, therefore, will be one of the numbers 0-6, or the least positive residues of modulus 7.

Keeping in mind the exercise we discussed above, the number of days between any two Sundays is always divisible by 7, no matter how many weeks intervene. Consequently, the interval of time between March 22 + d + E (Easter Sunday of the year we’re trying to determine) and any given Sunday in any previous year, will be divisible by 7. So if we begin with a definite Sunday, we can discover a general relationship for determining the date of Easter.

Gauss chose Sunday, March 21, 1700 as his Sunday reference date. Next, he determined a relationship for how many total days elapsed between March 21, 1700 and any subsequent Easter Sunday. That total would be 365 days times the number of elapsed years, plus the number of leap days in those elapsed years. (Remember every four years, has one leap day in it.) Again, this number will be divisible by 7, no matter how many years intervene.

If A is the year for which we want to determine the date of Easter, A-1700 will be the number of elapsed years. (For example, if we want to find Easter in the year 1787, then there were 87 elapsed years (1787-1700).

If we call i the total number of leap days, then the total number of days between Sunday March 21, 1700 and March 22 + d + E, for the year we’re investigating will be:

1 + d + E + i + 365(A-1700)

This number is divisible by 7, (because it is the number of intervening days from one Sunday to another).

At this point, the main conceptual problem has been solved. The date of Easter can be determined as March 22 + d + E, with d being determined by the calculation discussed last week, and E determined by the calculation which will be developed below.

Gauss was never content, unless he found the absolute simplest way to accomplish his task. All that remains is to simplify the above calculation so that E will be the residue which arises when the above number is divided by 7. Gauss accomplished this by repeatedly employing the principle, cited above, that adding or subtracting multiples of the modulus, doesn’t change the congruence. I include the following applications of this principle, even though it is expressed by some algebraic manipulations. The reader should focus on the addition and subtraction of multiples of the modulus 7.

To determine the number of leap days, i. we must first determine what relationship the year in question is to the leap year. Or, in the language of Gauss’ Higher Arithmetic, what is the least positive residue relative to modulus 4 of the year in question? This is the remainder b in Gauss’ algorithm. (For example, if the year is 1787, the least positive residue relative to modulus 4 is 3. That is, 1787 is three years after a leap year. So the total number of leap years between 1787 and 1700 is 87-3/4=21, or 1787-1700-3/4.)

In Gauss’ formula, the total number of leap days i will be:

1/4(A-b-1700)

If A is between 1700 or 1799. If A is between 1800 and 1899, then we have to subtract 1 because 1800 is not a leap year. For now, we will stick to the 18th century.

So the total number of days between March 21, 1700 and Easter Sunday in year A, will be:

1 + d + E + 365(A-1700) + 1/4(A-b-1700).

And this number must be divisible by 7.

This is pretty complicated and cumbersome. But as we know from Gauss’ Higher Arithmetic, if we add or subtract multiples of 7, the result will also be divisible by 7. So Gauss, through the following steps, adds or subtracts multiples of 7, in order to bring this unwieldy formula into a simple calculation.

First he adds the fraction 7/4(A-b-1700) to the above making: 1 + d + E + 365(A-1700) + 8/4(A-b-1700)

Multiplying all this out gives us: 1 + d + E + 367(A-1700)-2b which equals: 1 + d + E + 367A – 623,900 – 2b

Then Gauss subtracts 364(A-1700) (which is divisible by 7) which gives: d + E + 3A – 5099 – 2b

Then Gauss adds 5096 (which is divisible by 7) to get: d + E + 3A – 3 – 2b

Now Gauss eliminates any need for the reference date by replacing A in the following way. First, we divide the year by 7 and call the remainder c. That means, if we subtract c from the year, the result will be divisible by 7. Or, A-c will also be divisible by 7. In the next step, Gauss subtracts 3 times A-c or 3A-3c which gives: d + E + 3c – 3 – 2b

Finally Gauss subtracts this from 7c – 7d which gives: 3 + 2b + 4c + 6d – E.

Which means E is the remainder if we divide 3 + 2b + 4c + 6d by 7. So the determination of the Easter date is March 22 + d + E.

Unfortunately our work is not completely done. Because in the Gregorian calendar, not every century year is a leap year, the algorithm must change from century to century. Gauss also solved this problem using principles of Higher Arithmetic. We will take this up in future pedagogical discussions.

## Where Are You?

by Bruce Director

After spending the last three months determining where the asteroid Ceres is, it is more than appropriate to ask where you are? The question of determining one’s location, while elementary, is not exactly simple. The same principles underlying Gauss’ method for the determination of the orbit of Ceres, were applied by Gauss, in developing the principles of what he called Higher Geodesy. Over the coming weeks, we will begin to investigate Gauss’ discoveries in Geodesy, and the corresponding changes in thinking that are intertwined with it.

An apparent distinction between the astronomical problem of determining the orbit of Ceres, and the earth-bound question of determining one’s location, may have already arisen in your mind. In determining the orbit of Ceres, we were discovering the underlying motion that corresponded to the changes observed in the positions of the planet, from the earth, which itself was moving. On the other hand, in determining one’s location on the Earth, we are apparently determining a fixed point. Shouldn’t the determination of a fixed point, on the relatively stable earth, be a piece of cake, compared with the Ceres problem?

It should have become abundantly clear from the work on the Ceres orbit, that we can determine almost nothing by direct observation, or by some naive “yardstick” method of measurement. Yet, by discovering the underlying harmonic ordering of the universe, we can, by the principle of self-similar proportionality, take into our minds, distances, motions, and time intervals, that far exceed our physical capacities. Further, we can then make increasingly precise determinations of those distances, motions and time intervals.

Such determinations can only be made, provided we make the necessary changes in our way of thinking. For example, Gauss would never have been able to determine the orbit of Ceres, if he limited his thinking to Euler’s algebraic formalism, and Newton’s push-me, pull-me conceptions. Such a view would require collecting a set of relationships into the form of equations, lining up all the known quantities on one side, and all the unknown quantities on the other side. But this is an impossibility in the case of the Ceres orbit, as almost nothing about the orbit was known! Gauss, on the other hand, did not seek such a set of equations, but, instead, he developed a whole array of relationships, that reflected the harmonic ordering of the solar system. It is only in the interconnections of these relationships, not in any one, that the sought-for underlying harmonic ordering of the solar system in the smallest interval of action, is discovered. And that, only to one who is willing to make the necessary changes in thinking, and “read between the notes,” so to speak.

Or, another example to think about: How can we, in the short span of our temporal existence, make precise determinations of historical events centuries past, even when very little “information” about those events is known?

Consider the manner in which Nicholas of Cusa in {On Learned Ignorance} Book II, Chapter Twelve, discusses these issues:

“… It has already become evident to us that the earth is indeed moved, even though we do not perceive this to be the case. For we apprehend motion only through a certain comparison with something fixed. For example, if someone did not know that a body of water was flowing and did not see the shore while he was on a ship in the middle of the water, how would he recognize that the ship was being moved? And because of the fact that it would always seem to each person (whether he were on the earth, the sun, or another star) that he was at the “immovable” center, so to speak, and that all other things were moved: assuredly, it would always be the case that if he were on the sun, he would fix a set of poles in relation to himself; if on the earth, another set; on the moon, another; on Mars, another, and so on. Hence, the world-machine will have its center everywhere and its circumference nowhere, so to speak; for God, who is everywhere and nowhere, is its circumference and center.

“Moreover, the earth is not spherical as some have said; yet it tends toward sphericity, for the shape of the world is contracted in the world’s parts just as is [the world’s] motion.”

As Cusa indicates, the determination of any position on the Earth, as in the determination of a planetary orbit, requires the determination of the characteristic curvature in the small.

Some of the considerations are already known to you, if you’ve worked through Lyn’s rigorous pedagogy of Eratosthenes’ determination of the circumference of the Earth, along the meridian. Eratosthenes measured distance on the surface of the Earth, as a relationship of the angular change in the relative motion of the Sun and the Earth. That angular change, observed as a change in the length of a shadow, was translated into distance, measured as a segment of a great circle passing through the poles on the spherical Earth. In other words, a fixed distance on the surface of the Earth is measured as a function of the harmonic ordering of the solar system as a whole.

By Eratosthenes’ measurements, the Earth is a sphere, and its surface has a constant curvature in all directions. Thus, the distance corresponding to any given angular change, is the same no matter where on the surface, or in what direction that angle is measured. Consequently any location can be identified as the intesection of two great circles, such as circles of latitude and longitude. The distance between any two locations can be measured uniformly by the angular change along the great circle arc that joins them.

In 1620, the Dutch scientist Willebrord Snell, revived Eratosthenes’ methods with greater precision in his book {Eratosthenes Batavus}, and began to make measurements, not only of arcs, but of large triangular areas of the Earth’s surface. By the mid-1700s, measurements conducted at varying latitudes had determined that the distance on the Earth’s surface, corresponding to 1 degree of latitude, was greater in the higher latitudes than near the equator. These measurements confirmed what Cusa had indicated, that the Earth’s shape was not spherical, but was instead a surface of changing curvature. (Sir Isaac Newton, for perfectly fraudulent reasons, is credited with determining the non-sphericity of the Earth, a fraud perpetuated to the present day by British pagan science. Gauss’ geodesy and investigations into the anti-Euclidean nature of space-time completely destroyed this Newtonian fraud. In future pedagogical discussions, we will look into this matter in more detail.)

What does this mean for determining one’s location on the Earth? If the Earth is a sphere, then all equal arcs along a great circle on the Earth’s surface will measure the same physical distance, regardless of position. Conversely, if the curvature of the Earth’s surface is changing, then equal arcs along any great circle, will measure unequal distances, depending on the position of the arc.

An initial conception of this problem of measuring arc- length on a curved line of changing curvature, can be gained, by the now familiar comparison between the ellipse and the circle. (See Pedagogical Discussions #97196bmd001 and Part 6 of the Ceres series in NF 1/26/1998)

Since the arc-length of a curved line of changing curvature, such as an ellipse, cannot be measured by reference to simple angular change, we must measure the change in curvature itself. Leibniz and Huygens discovered methods of measuring such changes in curvature, by generating new curves, such as involutes and evolutes. Curves such as these can be described as “curves of curvature.” (Pierre Beaudry in Doc# 97526PB 001 has translated some important studies on this work. In future weeks, as we work through more examples of geodesical problems, the application of these curves will be explored in more detail.) Gauss and Kepler’s development of the relationships of the eccentric, mean, and true anomalies is another example of measuring the unmeasurable arc-length of a curve of changing curvature.

But these methods deal with the curvature of “curved lines.” A different problem arises when, for example, measuring an area on a surface of changing curvature, such as a large triangle on the surface of the Earth. The sides of such a triangle, are each sections of a great circle. Not only are we confronted with the problem of changing curvature on each great circle, but the type of change in curvature of the great circles differs from great circle to great circle, depending on the direction.

The following experiment may help illustrate the point:

Hard boil at least three eggs. Peel the shell off. Look at the changing curvature of the surface of the egg. Now cut one egg in half length-wise, another the short way, and the third at an angle. Look at the different curvatures of the cross-sections of each type of cut. Carefully trace each cross-section on a piece of paper and determine the center of each curve. Obviously, the arc-length of any portion of each curve depends on the position of that arc-length. Now re-construct an egg in your mind, with all three cuts. The arc-length of a segment along the surface of any one of the cuts depends not only on the position of that segment with respect to the cut, but also to the direction of the cut itself. Any small change in position and direction of the segment, will correspond to a change in the length of that segment.

As we will see, Gauss found a method to determine the area of surface of changing curvature, by changing his assumptions about the very nature of space itself, which he investigated in the seemingly unconnected domains of arithmetic, geometry, geodesy, geomagnetism, and astronomy. Our current trajectory will carry us into Gauss’ investigations into bi-quadratic residues, the theory of curved surfaces, and the development of the complex domains that are referenced by B. Riemann in his famous Habilitation paper. As Gauss wrote to his friend, the geodesist Hansen, “These investigations lead deeply into many others, I would even say, into the metaphysics of the theory of space, and it is only with great difficulty that I can tear myself away from the results that spring from it as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”

So far we have only dealt with a few of the geometrical considerations which present themselves in the investigation of geodesy. In future discussions we will take up a myriad of other geometrical and physical considerations that arise in these investigations. All this brings to mind the question: How fixed is one’s location on Earth anyway?

Don’t Lose Your Bearings

by Bruce Director

A musical tone, or even a series of tones, is at best ambiguous except in the context of an entire polyphonic composition. So to, a fixed position on the surface of the Earth is only determined in the context of the polyphony of the motion of the solar system and the motion of the human mind.

In carrying out the triangulation method described last week, we discovered, in principle, a means to measure a linear distance along the surface of the Earth, as a function of area. The area, in turn, was measured as a function of the angular change between the lines of sight connecting three locations. Those angles, in turn, have to be measured as a function of the curvature of the surface on which the area lies. The latter problem was identified, but a more elaborate investigation was left to a future pedagogical discussion.

A necessary first step, is to determine a means to fix the positions of the three locations which form the vertices of the triangle. In the domain of abstract geometry, the position of any location on an arbitrary surface, be it a sphere, ellipsoid, or a more irregular shape like a hard boiled egg, can be fixed with respect to an arbitrary frame of reference. But, when the surface is in motion, that arbitrariness disappears, as that motion is reflected in the nature of the singularities which arise on the surface.

On the surface of the Earth, all fixed locations can be determined with respect to the combination of the Earth’s motions, as those motions are reflected in the positions of certain singularities. First, the Earth’s motion around its axis gives rise to two singularities — the north and south poles — in relation to which all other locations can be determined. These are physically determined singularities, not abstract geometric ones. Their locations cannot be determined by any purely mathematical considerations. Purely mathematical considerations can only tell us that on a closed surface, two points can be related to each other as poles and antipodes.

But, for the Earth, the location of these poles can be found by the unique type of apparent motion of the Sun and the stars as observed from these points. For example, at the poles, the apparent motion of the Sun, is such that an observer standing on the pole would see the Sun rise at the equinox, move along an upward spiral path until the solstice, then move in a downward spiral path until the next equinox, when it disappears from the sky until the next equinox. Also, certain stars, or constellations, (such as Polaris, or the Southern Cross) remain in a relatively fixed position with respect to the north and south poles.

With the determination of the north and south poles, measured with respect to the Earth’s motion around its axis and around its orbit, all other points on the Earth’s surface can be determined. The two poles can be connected in the mind by an infinite number of closed curves. On the sphere, this would be a great circle. On an ellipsoid, this curve would be elliptical. On an irregular surface, such as an egg, the curve would reflect the curvature of the surface. Geographers call these closed curves meridians, or lines of longitude.

A position along any of these closed curves can be measured as an angular change from the observed position of the stars or motions of the Sun. For example, in the northern hemisphere of the Earth, in this era, the star Polaris is directly overhead at the north pole. Or, in other words, the line of sight to Polaris makes a right angle with the line of sight to the horizon at the north pole. As the observer moves south along the meridian, that angle decreases, until at the equator it becomes zero. Once in the southern hemisphere, the observer must look to the constellation called the southern cross. The angle that this constellation points to then increases until the observer reaches the south pole. Continuing along the same meridian, now northward, the angle of the southern cross decreases, until the observer reaches the equator, when again he sees Polaris. As he moves northward the angle at which Polaris is observed increases until, at the north pole, Polaris is again directly overhead.

Locations along all the meridians, at which the above observed angles are equal, can be connected by closed curves called latitudes. An observer’s latitude can thus be precisely determined by measuring the angle of the line of sight to the appropriate constellation with the horizon.

How, from the surface of the Earth, can we distinguish one meridian from another? Or, once we determine our latitude by the above method, how can we determine our longitude? Again, we must look at the combination of the Earth’s motions. Unlike the determination of change in latitude, which is perpendicular to the Earth’s motion, change in longitude is parallel to the Earth’s motion. Therefore, we must add to the observed position of the Sun or the star, the difference in time at which that observation is made. For example, while the angle of the line of sight to the Sun at its zenith, will be the same at two different longitudes which are on the same latitude, the Sun will reach its zenith at different times in each location. An observer at the more eastward position would measure the Sun reaching its zenith before the observer in the more westward position. The angular difference between these observations, which itself is a reflection of the Earth’s motion on its axis and its orbit, is therefore a determination of the difference in the longitude of the two positions. (In future discussions, we will investigate more fully the measurement of time.)

In this way, any location on the surface of the Earth, can be determined with respect to the poles, as a combination of angular changes of the observed positions of the heavenly bodies, whose positions reflect the harmonic ordering of the solar system as a whole. These positions can then be expressed precisely as the intersection of a curve of latitude and a curve of longitude.

As he made measurements of angular change for astronomical and geodetical determination, Gauss was prompted to question the metaphysics behind these measurements, which he presented as hints of in his second treatise on bi-quadratic residues.

On the surface of the Earth, any position is determined with respect to the directions north-south and east-west, which, as we’ve seen, are distinct relations determined by the motions of the Earth, on its axis and in its orbit. Any change in position on the surface of the Earth, implies a corresponding change with respect to both directions, that is, each position is doubly connected.

To better grasp this relationship, Gauss had to invent an entirely new polyphonic mathematical metaphor: the complex domain. Next week, we will take a closer look at that conception.

Keep Your Head Up: Complex Motion is Simple

by Bruce Director

In last week’s discussion, we demonstrated that a position on the surface of the Earth can only be determined as the intersection of the various motions of the Earth itself. These motions are reflected in the physical geometry of the Earth’s surface as latitude and longitude, and such singularities as the poles and the equator.

To restate the irony: To determine a stationary position on the surface of the Earth, you measure the observed motion of heavenly bodies such as the Sun and the stars. But, these heavenly bodies are themselves relatively stationary, and it is the Earth which is moving, both on its axis and in its orbit. That movement of the Earth is observed as an angular change in the positions of the heavenly bodies. That angular change, itself changes, with respect to different positions on the Earth’s surface.

All positions on the same latitude, observe the motions of the heavenly bodies, such as the North Star, Southern Cross, or the angle of the arc of the Sun, from the same angle. All positions along the same longitude will observe those phenomena at different angles, but at the same time. For example, the observed angle of the Sun at its zenith will vary along a circle of longitude, but all positions along that circle will observe the zenith at the same time. (We measure time here, not by a clock, but as an angular change. For example, when the Sun reaches its zenith along one circle of longitude, the Sun will be past its zenith along circles of longitude to the east and before its zenith along circles of longitude to the west. The angle of difference either pre- or post-zenith, is our measure of time.)

In the language of Gauss’ higher arithmetic, the astronomical observations from all positions on the same latitude will be congruent relative to that latitude as a modulus. Similarly, the astronomical observations from all positions on the same longitude are congruent relative to that longitude as a modulus. The moduli, in this case, are not simple numbers, they are distinct physical phenomena.

Furthermore, on the surface of the Earth, any position is determined as the combination of both moduli at once. Any change in position on the surface of the Earth, simultaneously reflects a change in those astronomical orientations which determine both latitude and longitude. Or, in the language of Gauss’ higher arithmetic, the positions on the Earth are congruent relative to a complex modulus.

A major impediment to grasping this physical concept, is the impulse, a “bad habit,” to falsely believe you are squatting outside the Earth, and looking down on a spheroid which has a Cartesian net drawn on it. In this virtual reality, position on the Earth’s surface is defined simply as the intersection of two circles or lines on a map or globe. This conceptual difficulty will persist, unless you, at least for now, throw away the map. Reality lies not in the map, but in the conceptions from which the map is made. As you work through this, you should happily find and root out the remnants of Cartesian-Kantian notions of space-time, which trap your mind in virtual reality.

Gauss himself stated that these concepts, while clear in his mind, become only a vague picture when he tried to put them in words. The best approach is to provoke the mind out of the Cartesian-Kantian dogma, through a series of seemingly unrelated paradoxes. If you can keep several paradoxes active in your mind at one time, the corresponding cognitive process should give rise in your mind, to a sharp conception.

Think again on the question of changing position on the surface of the Earth. In any change in position, you are not moving independently in a north-south direction, and then in an east-west direction as implied by a Cartesian-Kantian map. Your change in position is in both directions simultaneously. Any small change in position, reflects a corresponding change in the astronomical observations we have used to determine the latitude and longitude of that position.

Another example is from Gauss’ “Questions concerning the metaphysics of Complex Numbers.” (See Fidelio Winter 1997 p. 105) The position of the bubble in a plane leveler moves back and forth only when combined with the up down motion of the ends of the level. Here motion in the horizontal direction is not independent of the vertical but combined with it.

Abandon all preconceptions of space and time and join the following journey. Starting at the equator, we observe the North Star just on the horizon. As we move northward along the meridian, the observed angle of the North Star increases from zero until, when we reach the North Pole, that observed angle is 90 degres. As we continue along the same meridian — we are now moving southward — the observed angle of the North Star now decreases from 90 degrees to zero when we again reach the equator, on the opposite side of the Earth from our starting point. Each increase of distance (length) along the Earth’s surface, corresponds to an increase or decrease in the angle of observation of the north star. In fact, we measure the increase in distance (length) by a change in that angle.

Still keeping away from the Cartesian-Kantian conceptions of space-time, what geometrical metaphor arises in the mind from the doubly connected action encountered in the above journey?

Two different {types} of change occur as we move along the meridian. A change in length and a change in angle. The distinction between adding lengths and adding angles, in classical Greek geometry is identical to the distinction between an arithmetic and geometric progression. When lengths are added an arithmetic progression results. When areas or angles are added a geometric progression results. (See Geometric Numbers: The Prisoner and the Professor #97326bmd002.)

Classical Greek geometry recognizes that regardless of the increment added, each progression can be characterized by the nature of its mean, or what happens when the progression is divided. When a length is divided, a commensurable magnitude results. When an area is divided by an angular change, an incommensurable magnitude arises, such as when a square whose area is one is divided along the diagonal forming the square root of 2. The arithmetic mean is calculated as half the sum between the extremes. The geometric mean is calculated as the square root of the product of the extremes.

Gauss investigated this concept in a new way in the beginning of his Disquisitiones Arithmeticae. Take an arithmetic and geometric progression, and form the period of least residues with respect to the same modulus. Two entirely different orderings arise. (See Beyond Counting #97396BMD001; 97406BMD01.)

Now back to the surface of the Earth. Our journey was simultaneously an arithmetic progression (change in length) and a geometric progression (change in angle). The North Pole was half the length of our journey along the Earth’s surface, i.e., the arithmetic mean, while the angle of observation of the North Star was at half a rotation, the geometric mean. In terms of the geometry of our journey, the North Pole is the arithmetic- geometric mean.

Now in the metaphor of numbers. From the standpoint of the North Pole, where the angle of observation of the North Star is 90 degres, we can move in one of two directions along the meridian to reduce that angle (either towards our starting point, or towards the ending point). We can distinguish one direction as negative and the other direction as positive. Assigning magnitudes to our metaphor, we can then think of the North Pole as 0, which is the arithmetic mean between our starting point -1, and our ending point +1. And, the North Pole is also the geometric mean, between the angles of observation of the North Star, or in numbers, the square root of -1 x +1 — the square root of -1. Any change on our doubly-connected journey can be expressed as an arithmetic component (length) plus a geometric component (angle). Giving the geometric component the symbol i, this doubly-connected action is represented as a difference between two complex numbers of the form a + bi.

The astronomical ordering of positions on the surface of the Earth, latitude and longitude, is similarly a doubly-connected complex domain, as all changes in position have both an arithmetic (change in length) and geometric (change in angle) component. Remember that the measurements made in a triangulation, had both these components. In fact, the doubly-connected manifold represented by latitude and longitude, is actually the reflection of a higher multiply-connected manifold of the various cycles of the Earth’s motion.

This is not the formal Cartesian-Cauchy concept of the “complex plane” you probably reviled in school, but the “fully alive” metaphor of the complex domain elaborated by Gauss in his Theory of bi-quadratic residues and the Investigations into the nature of curved surfaces. The polyphonic character of this metaphor will emerge as we investigate the geometry of the complex domain. (A glimpse of this polyphony was explored in the very first series of pedagogical discussions on prime numbers.)

The words of Gauss in his letter to Hansen are worth recollecting at this point: “These investigations lead deeply into many others, I would even say, into the metaphysics of the theory of space, and it is only with great difficulty that I can tear myself away from the results that spring from it as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”

That vague image will become sharper as we delve further into Gauss’ concept of higher geodesy.

Bi-Quadratic Residues: How to Change the Way You Think

by Bruce Director

A curious thing happened two weeks ago as we took a journey along the meridian, from the equator, through the North Pole, and on to the equator again. We confronted the reality that our change in position on the Earth, is inseparably connected to a change in the angle of observation of the North Star. The Cartesian/Kantian notion that space is three infinitely extended independent directions, was shown to be an illusion.

Had we conducted this trip with our head to the ground, like a Baby Boomer in a bad feeling state, we wouldn’t have discovered any of this. Instead, we would have measured our progress as simply a linear change of distance along the ground. For a Baby Boomer in a feeling state, this would have been quite comfortable, since nothing in the journey, no matter how physically arduous, would disturb those Cartesian/Kantian axioms of space as linearly extended.

By lifting our eyes to the stars, and keeping them there as we move, we changed the way we think. Now, we should tune our minds, as Gauss did, through the polyphonic metaphor of arithmetic and geometry, so that our heads never sink to the ground again.

From the standpoint of abstract geometry, our journey could be represented as a semi-circle. The motion of the North Star represented by the circumference, while our motion along the Earth’s surface represented along the diameter. However, such an abstract representation leaves out the crucial physical singularity of the North Pole. Not only is the North Pole the maximum angle of observation, but also, at that position, the direction at which we observe that angle suddenly changes. As we move northward from the equator, the angle increases in front of us. After we pass the North Pole, that angle decreases {behind} us. We have to turn opposite to the direction of motion, to see the North Star.

As Gauss wrote to C. L. Gerling on Feb. 8, 1846: “The difference between right and left is not capable of definition, but only of demonstration, as is similarly the case with sweet and bitter. But all simile limps. The latter values have a reality only for the taste buds, but the former for all minds for which the material world is apprehensible. But, two such minds cannot directly come to an understanding about right and left, except as one and the same individual material thing builds a bridge between them. I say directly that A can communicate with Z, when A builds, or is able to build, a material bridge between A and B, another between B and C, etc. What worth this matter has for metaphysics, I articulated succinctly in the (announcement of the second treatise on bi-quadratic residues), and in it I’ve found a conclusive refutation of Kant’s illusion, that space is MERELY the form of our external perception.”

Now turn to the surface of the Earth. In previous discussions, we showed that the concepts of latitude and longitude, by which we determined positions on the Earth’s surface, actually reflect the orientation of those positions with respect to the Earth’s motion. The multiply-connected motions form, on the Earth, a multiply connected-surface. Now, let’s investigate the relationship between several such positions.

This touches on the subject of Lazare Carnot’s famous study, “Geometry of Position,” as well as the work of Desaurges, Monge, and Poncelet, which were all extensions of Leibniz’ Analysis Situs. Gauss was already familiar with Carnot’s investigations, but in 1810, when his student and friend, H.C. Schumacher translated Carnot into German, Gauss submitted a re-working of some of Carnot’s discoveries. The eighth volume of Gauss’ collected works, contains a series of fragments on the application of complex numbers to geometry, which were found as annotations in Gauss’ copy of Schumacher’s translation.

Under the Geometry of Position, we investigate the geometrical relationships among several positions, which, as we’ll see, are a function of the domain in which those positions exist. But, as in the case of the surface of the Earth, or the orbit of an asteroid, the nature of that domain is not discernable directly, but only from the relationships among the positions. Only by investigating how those relationships change, when pressed to the limit under the Socratic principle of exhaustion, can we provoke a paradox, whose solution gives rise to an insight into the nature of the domain itself.

For example, take what may appear to be the simplest case, which we have already touched on in a previous pedagogical discussion. Determine the latitude and longitude of one position A, by astronomical measurements. Now, determine, by astronomical measurements, the latitude and longitude of another position B. To keep the case simple, choose B along the same meridian, but at a different latitude, say 1 degree north from A. In other words, the change in the angle of observation of the North Star from A to B, will be 1/90 the total change of that angle from the equator to the pole, while the zenith of the Sun will be observed at the same time at both positions. Is the distance from A to B 1/90 the distance from the equator to the pole, along the surface of the Earth?

The answer to this question depends on the curvature of the Earth. If the Earth is a surface of constant curvature, such as a sphere, then the angular change between A and B, and the distance relationship are the same, regardless of the position of A and B. If, on the other hand, the Earth is a surface of changing curvature, then the distance relationship between A and B, will depend on the position of A and B with respect to the physical singularities of the Earth’s surface. (Don’t forget that those singularities, the poles and equator, are themselves a function of the Earth’s motion. So, the distance relationship between A and B depends not only on their geometrical position on the Earth’s surface, but also their relationship to the Earth’s motion.)

The rub seems to be, that the determination of the curvature of the Earth, can only be made by measuring the distance between A and B at several different places, and then looking at the change. But, an accurate measurement of the distance between A and B, at any location, depends on knowledge of the curvature of the Earth at that place. We cannot make an accurate determination of the length of a curve, unless we first determine the curvature, since that curvature will affect our measurements at every interval no matter how small. Furthermore, if the Earth is a surface of non-constant curvature, you would have to measure an arc at a large number of different locations, before getting any competent idea of the nature of that surface.

This empirical method, of determining the Earth’s shape by making arc measurements at different locations, was the subject of a major political fight, the history of which merits further investigation. Working in France, around the turn of the 18th Century, the Cassinis measured several arc lengths. According to their results, the distance between 1 degree of latitude got shorter as they moved north. This variation indicated that the Earth was a prolate ellipsoid (that is an ellipse rotated around its major axis). The Cassini results contradicted Newton’s speculations that the Earth was oblate (that is, an ellipsoid rotated around its minor axis.) But Newton’s speculations were fraudulent. Huygens had earlier proposed that the Earth was “flatter” at the poles, based on his investigations into the pendulum, and observations that a pendulum at the equator lost 2 minutes 28 seconds a day. In their zeal to prove Newton right, the oligarchy deployed Maupertuis and others to measure arcs in Peru and Lapland. The Lapland arcs were longer, indicating the shape was an oblate ellipsoid generating manic elation among the Newtonians. Voltaire pronounced, “Maupertuis has flattened the Earth and Cassini!” Of the Peruvian measurements, Voltaire said, “You have found by prolonged toil, what Newton had found without even leaving his home.” Maupertuis had an egotistical picture drawn of himself, flattening the Earth with his foot!

Despite Voltaire’s frothings, nothing was actually proved by these measurements. The Earth’s surface, while closer to an oblate than a prolate ellipsoid, is a much more complicated dynamic surface — a surface of non-linear, non-constant curvature. Such a surface cannot be approximated by any simple geometrical shape, as attempted by Newton and Maupertuis.

Instead of thinking of simply measuring distances, we should think of measuring changes in curvature. Or, in other words, with a constant change of latitude (1 degree) there will be a corresponding non-linear, non-constant change in the distance between two positions A and B. The difference, between the constant change of latitude, and the non-constant change of distance is very small, but it is in that small change, that the curvature of the Earth can be determined. That is the small change, which Gauss set out to measure.

In 1821-1824, Gauss undertook, as part of his triangulation of Hannover, to measure the arc along the meridian between the his observatory at Goettingen and Schumacher’s at Altona. Both men made observations of the zenith at the same time and determined that the observatories were on the same longitude. Gauss set about to measure the distance along the surface of the Earth between them, to determine the difference between the astronomical and their geodetic positions. His aim was not simply to measure the distance between the observatories, but to develop a method to measure non-linear, non-constant curvature, establishing a new revolutionary universal principle.

To begin the measurement, Gauss formed his first triangle between Goettingen, Hohenbogen, and Hils, which were all locations visible from one another. Once located, he measured the angles between the lines of sight connecting the three locations.

Before going any further, we must take notice of an underlying problem inherent in the measurement of physical angles. The astronomical angles are vertical angles with respect to the horizontal. In order to measure these angles you have to first determine the horizontal. This is done by levelling the measuring instrument with a spirit bubble, so that when the instrument is parallel to the horizontal, the bubble in the tube is centered. Determined in this way, horizontal is not an abstract geometrical property, but a physical relationship between the measuring instrument and the Earth’s gravitational field. What if, as was already determined by previous observations, the gravitational field of the Earth was not constant? These gravity anomalies would thus affect the determination of the horizontal from which these angles were measured and, what we call horizontal itself changes with position.

Furthermore, while the lines of sight between Goettingen, Hohenbogen, and Hils are straight lines, the sides of the surface triangle formed by these points, are arcs of curves. If the Earth is a sphere, then these are arcs of great circles. If the Earth is an ellipsoid, than each arc is different. If the Earth is more irregularly shaped, as it is, the curvature of each arc is non-constant in the small.

So the relationship between these three positions, is a function of the curvature of the surface on which they lie, precisely what Gauss is trying to determine! We will pick up next week on this problem, but it is useful to remember what Gauss said in 1809:

“Everything becomes much simpler, if at first we abstract from the infinity of divisibility, and consider merely discrete magnitudes. For example, as in the biquadratic residues, points as objects, as transitions, and hence relations as magnitudes, where the meaning of a + bi – c – di is immediately clear.”

TRIANGLES

by Bruce Director

Last week we began to re-create the discovery of principles underlying Gauss’ determination of the shape of the Earth.

Gauss and Schumacher had both made very accurate astronomical measurements of the angle of inclination of observation of the North Star from the observatories at Goettingen and Altona. They also made accurate measurements of the difference in time at which particular stars crossed the zenith at each location. These measurements provided Gauss with a very precise determination of the astronomical latitude and longitude of the two observatories. Or, as we have emphasized throughout this series of pedagogical discussions, the orientation of these two positions with respect to the motions of the Earth.

These measurements determined that both observatories were on the same meridian. The angle of observation of the North Star from Goettingen was: 51 degrees 31 minutes 47.85 seconds. At Altona, the measurement was: 53 degrees 32 minutes 45.27 seconds. The longitude for both observatories was determined to be the same, because, any given star could be observed directly overhead at the same time, in both places.

Gauss now set out to determine the relationship between these positions, with respect to the surface of the Earth. Should there be a difference? This is not a self-evident question, since astronomical latitude and longitude changes with respect to the position of the observer on Earth’s surface. Under what conditions would there be a difference between the astronomical and geodetic latitude and longitude? And, how could such a difference be measured?

Since the two observatories had the same longitude, they were on the same great “circle”, going through the North and South poles, separated by 2 degrees 0 minutes 57.42 seconds, or approximately 178th the circumference of that meridian. If the Earth were a perfect sphere, and the great “circles” were truly circles, then the physical distance between the two observatories could be computed by dividing 178 into the circumference of the great “circle” on which they lie. But as we’ve discussed before, if the great “circle” were an ellipse, a different calculation would result, depending on the eccentricity and length of the major axis of that ellipse. If, on the other hand, that great “circle” were a more irregular curve, an entirely different method would be required to calculate the physical difference between the two latitudes. Since the curve on which the two observatories lie, is not necessarily a circle, we will use the more general term, {geodesic}, which describes the shortest curve connecting two points on a surface.

So the determination of the physical difference between these two locations, from the astronomical latitude and longitude, depends on an assumed shape of the Earth along that meridian. But, on what basis can such an assumption as to the shape of the Earth be made? A precise determination can only be made by measuring the change in curvature along the Earth’s surface. This appears to be a vicious cycle. To measure the distance requires an assumption of the shape of the Earth which can only be determined by the measurement itself!

To measure the difference between these positions with respect to the Earth’s surface, Gauss applied the method of triangulation, albeit more refined, that was developed by the Dutch republican, Willibrord Snell. This method consisted of finding locations which could be seen from each other and measuring the angles between the lines of sight connecting the locations. Once these angles were measured, the methods of trigonometry, (extensions of the principle of proportionality of self-similar triangles) can be employed to determine the distance between the points.

To be concrete, Gauss began with a triangle between Goettingen, Hohenbogen, and Hils. All three locations were on hills and had good views of each other. Hohenbogen, had a signal tower on it that had been built by the French. Gauss had to construct one himself at Hils.

The first step was to determine the distance between Goettingen and Hohenbogen. This was accomplished by measuring an intermediate, smaller triangle, formed by the observatory at Goettingen, the signal tower at Hohenbogen, and a meridian marker placed by the French, approximately 2 km from the Goettingen observatory, on, as the name implies, the same meridian. The distance from the meridian marker to the observatory had been measured by a French triangulation, and was checked by Gauss.

At this point, Gauss was using a standard theodolite, which is a telescope mounted on circular base, so that telescope can be rotated around the circular base, and also perpendicular to it. Later he would invent a superior device, which he called the heliotrope.

While one of his assistants positioned himself at the meridian marker, Gauss, standing at the Goettingen observatory, leveled his theodilite to make it perpendicular to the pull of gravity, and aimed the telescope at the meridian marker. Then Gauss rotated the telescope around the circular base until he could see his other assistant standing at the tower at Hohenbogen. The arc of the circular base between the position of the telescope when he was looking at the meridian marker, and the position of the telescope when he was looking at Hohenbogen, was the angle between these two sides of the triangle.

Then Gauss repeated this process from the signal tower at Hohenbogen, taking careful aim at the observatory at Goettingen, and then rotating the telescope to see the meridian marker. The circular arc formed between these two positions, was the measure of this angle of the triangle.

Repeating this process from the meridian marker, gave Gauss a measurement of the third angle of the triangle.

Now Gauss had measured three angles of a triangle, and he knew from previous measurements the length of the side between Goettingen and the meridian marker.

Were this triangle on a plane, then Gauss could apply some basic principles of Euclidean geometry, specifically, the proposition that if two triangles have the same angles, their corresponding sides will be proportional. Then, applying this principle using trigonometric relationships, he is able to calculate the appropriate distances. The distance from Goettingen to Hohenbogen (G-H), could be calculated as the distance between Goettingen and the meridian marker (G-M) divided by the sine of the angle at the meridian marker (m) divided by the sine of the angle at Hohenbogen (h). Or, in symbolic terms, G-H = G-M x (sine m/sine h). (The reader is advised to make a sketch of this triangle labeling the sides and angles accordingly.)

Simple, eh? Not so fast! The proposition of similar triangles is based on another principle of Euclidean geometry, that is, that the sum of the angles of a triangle is always equal to two right angles, or 180 degrees. And, this proposition is in turn based on the famous parallel postulate of Euclid.

We know, however, that this triangle is not a plane triangle, because it is a triangle between three positions on the surface of the Earth which we know, by Eratosthenes’ experiment, to be curved. To take a preliminary stab at the significance of this, consider this surface to be spherical. A simple observation of the nature of a spherical surface, shows us that the sum of the angles of a triangle is not always equal to 180 degrees. In fact, the sum of the angles of a spherical triangle is always greater than 180 degrees and can be as large as 270 degrees. (Consider the triangle formed by the equator and two meridians directly opposite each other). Is there some principle of similarity with respect to spherical triangles, as there is with respect to plane triangles?

This is a very rich and bountiful subject for investigation, which will be developed more fully in a series of future pedagogicals. The reader is initially referred to the 21st Century article on Gauss’ work on the Pentagrama Mirificum. For purposes of this discussion, we will introduce only some preliminary considerations.

Unlike a plane triangle, a spherical one is made up completely of angles. The sides of the spherical triangle, are arcs of circles whose center is the center of a sphere. Therefore the side of a spherical triangle can be expressed as the angle between the two radii connecting the center of the sphere to the endpoints of that side. The sides, in turn, form angles between each other. Thus, the spherical triangle defines a solid pyramid, whose base is the triangular section on the surface of the sphere, and whose apex is the center of the sphere. A series of relationships among these six angles can be developed. The exact relationships are not necessary for this discussion, but will be developed in the future.

But, what good is any of this for the problem of determining the distance between Hohenbogen and Goettingen? To utilize the principles of similarity of either plane or spherical triangles, we first have to make an assumption about what type of surface the triangle we are measuring is on. The wrong assumption, and the determination of the distance is wrong.

The problem here is a purely subjective one. Our problem is not which geometry, plane or spherical we should apply. But, more fundamental, our problem stems from being subjectively locked into making an a priori determination of which geometry to apply. That is, the characteristics of the surface determine what geometrical relationships should apply to our distance measurement. But those characteristics must be determined from our measurements, not a priori. To determine these characteristics, we must utilize the concepts developed in Gauss’ treatment of bi-quadratic residues.

Long before Gauss ever made his first geodetic or astronomical measurement, he had begun to re-consider the entire foundations of geometry. As early as 1792, when he was only 15 years old, Gauss realized that which geometry is the “real” one cannot be determined, as Kant insisted, a priori, but must be demonstrated.

For example, in an 1829 letter to Bessel, Gauss says: “Also about another theme, that for me is almost 40 years old, and which I think about again from time to time in my free hours. I mean the fundamental basis of Geometry: I don’t know, whether I’ve ever spoken with you about my views. I also have to consolidate many of them further, and my conviction the we are not able to prove Geometry completely a priori, is, where possible, still firm. Meanwhile, I still have a long way to go, to work out a published announcement of my extended investigations about it, and it may never happen in my lifetime, as I fear the shrieking of the Boeotiers, if I were to completely express my views. (Boeotiers was a term for the uneducated rabble that Gauss and his friends used as students.) But it is odd, that outside the well known deficiencies in Euclid’s Geometry, which, to the present time, only a gratuitous search has been made to fill out, there is yet another defect, which to my knowledge no one to the present day has attacked, and there is no way (though possible) to easily remedy it. This definition of a plane as a surface in which any two points are connected entirely by a straight line. This definition contains more than is necessary to determine the surface, and tacitly involves a theorem that first must be proven…”

Next week, we will start to look into the principles Gauss applied to determine the nature of the surface without a priori assumptions. For starters, think about something which probably skipped past you at the beginning of this discussion. In order to make his measurements, Gauss had to first level his theodilite so that it was perpendicular to the pull of gravity. What direction is that?

WHAT’S UP

by Bruce Director

Last week, in our efforts to re-trace Gauss’ determination of the difference in latitude between the observatories at Goettingen and Altona, we began to confront the subjective considerations, which underlay any such measurements.

To review the problem: Gauss and Schumacher have made precise astronomical measurements of the latitude and longitude of the two observatories, by measuring the angle of inclination of the North Star, at the two locations, and determining the time at which certain stars appear to cross the meridian at each location. This yielded the astronomical measurement that both observatories were on the same meridian, with Goettingen being at 51 degrees 31 minutes 47.85 seconds N. latitude, and Altona being at 53 degrees 32 minutes 45.27 seconds N. latitude, a difference of 2 degrees 0 minutes 57.42 seconds. These measurements are, as previously discussed, the orientation of these two locations with respect to the multiply-connected cycles which make up the Earth’s motion in the solar system.

Gauss, as part of his overall task of measuring the Kingdom of Hanover by a triangulation, set out to determine the distance as measured along the surface of the Earth, between these two locations. This led immediately to a paradoxical situation because the measurement of distances along the Earth’s surface <seemed> to depend on prior knowledge of the shape of that surface. The ultimate distance along the meridian between the two observatories would vary, depending on the type of curve that connected them, which in turn depended on the overall shape of the surface. For example, if the Earth were a sphere, then the arc between the two observatories would be a section of a circle; were the Earth an ellipsoid, the arc would be a section of an ellipse; or, were the Earth something even more irregular, the curve connecting the two locations would be something entirely different. To measure the distance between the observatories, thus requires knowledge of the type of curvature of the arc.

We seem to be stuck in the paradox of either measuring the distance linearly along the surface and then finding a curve onto which that distance “fits,” or, making an a priori assumption about what the curvature of the surface is, and interpreting our measurements from that standpoint.

As indicated at the end of last week’s discussion, this paradox confronts us, even before we measure the first angle of our triangulation, or the astronomical observations. These angles are measured by using a telescope which is mounted so as to be able to rotate in both a vertical and horizontal direction. The angles of inclination of the stars are measured as the angle between the horizontal base and the line of sight to the star. The angles of the triangles are measured as the amount the telescope had to be rotated, around the circular base, when viewing two of the vertices of the triangle from the third.

In order for any of these angular measurements to mean anything, we first must determine a way to fix what we call horizontal.

This might seem like a simple question. One might say that we can equate horizontal with the visible horizon. This would be unacceptable, not only because the visual horizon is not very precise, but how would one precisely determine the horizontal if you were standing in a forest, or in a valley surrounded by mountains?

From at least the time of the ancient Egyptian surveyors, the horizontal direction was determined as the direction perpendicular to direction of a hanging string with a weight on it, called in Greek a gnomon, from the word to know. This direction is the direction of the pull of gravity, and is not always perpendicular to the ground, as, if, for example, you were standing on the side of a hill.

By attaching the string to the center of the circular base of the telescope, the base can be positioned so as to be perpendicular to the string. When this is done, the angles made by rotating the telescope will be changes from a uniformly determined horizontal. An even more precise means of determining the horizontal, is to use a spirit level. This is simply a glass tube containing a small amount of alcohol and a bubble of air. Two circles are drawn on the tube that are perpendicular to the axis of the tube. The bubble will move back and forth in the tube, depending on the angle of the level with respect to the direction of the gnomon. When the level is perpendicular to the direction of the gnomon, it floats between these two circles, and is thus horizontal.

With this means of determining horizontal, we can move the telescope from place to place, measuring both vertical and horizontal angles. In this way, we can know, for example, that when Gauss and Schummacher measured the angle of inclination of the North Star at two different locations, that those angles were comparable. Or, when Gauss measured the angles between the lines of sight connecting Goettingen, the meridian marker, and Hohenbogen, that those angles were all in the same plane.

Once again, however, there is an underlying assumption which cannot be ignored. The angle between the level and the gnomon remains the same (90 degrees) everywhere, but what if the gnomon doesn’t always point in the same way? Or, in other words, does the curvature of the surface of the Earth affect in some way this determination of the horizontal? If so, then we are confronting again the very same paradox: The determination of the curvature of the Earth depends on measurements, which in turn are affected by that curvature.

What we call vertical and horizontal with respect at any point on a curved surface can be defined geometrically as the normal and the tangent to the surface at that point. For example, on a sphere, the horizontal direction is the direction of the tangent to the sphere at that point. The vertical direction is perpendicular to the tangent, or normal to the surface at that point. All normals to the sphere, when extended inward, will intersect at the center of the sphere.

But, on a surface of changing curvature, such as an ellipsoid, an anomaly arises. If we define vertical and horizontal geometrically as we did in the sphere, this direction is different for every point on the surface.

To illustrate this for yourself, experiment again with a hardboiled egg. Peel the egg, and place a toothpick through the short axis of the egg, as close to the center as you can, so that it comes out directly on the opposite side of the egg. This toothpick corresponds roughly to the Earth’s axis of rotation, and the two places where the toothpick sticks out of the egg shall correspond roughly to the North and South Poles. Then take another toothpick and stick it into a point close to what would be the egg’s equator, so that the toothpick is normal to the egg’s surface at that point. Push this toothpick part way into the egg. Now stick a third toothpick part way into the egg, at a point somewhere between the other two, so that it is also normal to the egg’s surface. Now, slice the egg in half along the long axis, being careful not to disturb the toothpicks. Once cut, if you push the second and third toothpick through the egg carefully, you will see that they intersect the first toothpick at different points.

Now consider the implications of this geometrical characteristic for geodetic measurements. When the vertical is determined by the direction of the gnomon, that direction points towards the center of the Earth. Were the Earth a sphere, the direction of the gnomon from two different locations, would coincide with the direction of the normals to the Earth’s surface. But, on an ellipsoid, the gnomon would still point to the center of the Earth, its direction being determined by the pull of gravity, while the direction of the normal, as in the case of the egg, would be different for different points on the surface.

Here physical and abstract geometry diverge. In abstract geometry, the shortest line connecting two points is known as a geodesic. On a plane, this curve is a straight line. On a sphere this curve is a great circle. On an ellipsoid, this curve is an ellipse. But, on the physical surface of the Earth, the concept of geodesic has a significant difference. If we level a telescope at a point A, and sight to another point B, which is farther north, the line of sight will trace out a curve along the surface of the Earth. If we then move the telescope to B, level it and sight to A, the line of sight will also trace out a curve on the surface of the Earth. Are these curves from A to B and B to A, the same?

Were the Earth a sphere, the two curves would coincide, as the vertical, determined by the gnomon, at both points would be in the same direction as the normals to the surface. And, both normals would intersect at the center of the Earth, the direction toward which the gnomon points. The curve along the surface of the Earth connecting A and B would coincide with a great circle, or geodesic.

However, were the Earth an ellipsoid, the direction of the normals at A and B would be different than the direction toward the center of the Earth, as we illustrated with the egg. That means that the horizontal would be pointing in a different direction for each point. So, if we positioned the telescope at A, leveled it and pointed it at B, the curve that the line of sight traced out on the surface would be different than when we positioned the telescope at B and pointed it back at A. The actual geodesic between A and B, would not be a simple ellipse, but the curve traced out, if we were to move the telescope from A to B, keeping B in our sight at all times, and also keeping the vertical plane of the telescope normal to the ellipsoid.

And that’s not all. Careful measurements have determined that the direction of the pull of gravity on the surface of the Earth is not always towards the center, but varies with the topography. This phenomenon is known as “deflection of the vertical,” which changes the vertical more irregularly than even an ellipsoid.

All our astronomical measurements have to now be reconsidered. They can not be considered simply the angle between the horizon and the inclination of the line of sight to the star we are observing. Rather, these are angles between the line of sight and the direction of the gnomon, which is not the same everywhere on the Earth.

Since all our measurements are determined by the direction of the gnomon, and this direction changes virtually everywhere with respect to the physical surface of the Earth, this raises the question, “What surface are we measuring when we measure the surface of the Earth?”

In his essay on the “Determination of the Difference in Latitude Between the Observatories at Goettingen and Altona,” Gauss stated, “What we call the surface of the earth in the geometrical sense is nothing more than that surface which intersects everywhere the pull of gravity at right angles, and part of which coincides with the surface of the oceans.”

This surface became known as the Geoid, and it is a far different surface than either the physical surface of the Earth, or an ideal geometrical shape such as an ellipsoid. The Geoid is such an irregular surface that it deviates everywhere from an ellipsoid, which in turn deviates from the surface determined by astronomical measurements. With this in mind, we can no longer think of the surface of the Earth, but instead the surfaces of the Earth. Or, alternatively the Earth must be considered as a multiply-connected surface.

So, by undertaking to measure the distance between two locations on the Earth’s surface by triangulation, Gauss was in effect determining the interaction between the multiply-connected cycles of the Earth’s motion, and the multiple curvatures of the Earth’s surface.

Surfaces and Triangles

by Bruce Director

A major subjective breakthrough was achieved with Gauss’ introduction of the concept that the geometrical surface of the Earth is the surface that is everywhere perpendicular to the pull of gravity, called the Geoid. Under this conception, Gauss applied his longstanding conviction, that geometry of physical space-time, cannot be determined by any a priori considerations, but is a matter to be determined by measurement, which is a function only of cognition. The breakthrough, is not that Gauss somehow discovered the “true” shape of the surface of the Earth, but that Gauss discovered the quality of mind from which that shape can be so determined.

Review the implications of this for the previous matters we’ve discussed in this series of pedagogicals. As we’ve seen, every point on the surface of the Earth can be determined as a precise intersection of the various cycles of the Earth’s motion. The latitude, for example, is determined by the angle of inclination of the line of sight by which we observe the North Star. That angle is, itself, a combination of the Earth’s position in its orbit, the longer cycle of the precession of the equinox, and the intermediate variations in the precession, known as nutation. These factors, and others, make the North Star’s position vary with respect to the Earth’s north pole. That is, the North Star is not always exactly true north. Latitude, therefore, is determined not only with respect to the angle of inclination of the North Star, but also the angular change associated with precession and nutation.

As we now know, the observed angle of inclination of the North Star, is actually the angle between the line of sight of the North Star, and the direction of the plumb line. It is a relationship between two changing surfaces, the Geoid and the astronomical cycles mentioned above. This relationship is called the astronomical latitude.

The same point on the Earth’s surface has a different latitude with respect to surface and center of the Earth. Geodetic latitude is the angular change along the surface of the Earth, with respect to the center of the Earth itself. Geodetic latitude is measured as the angle between the normal to the surface of the Earth, projected inward, and a line in the plane of the equator. This line would be perpendicular to the Earth’s axis of rotation and go through the center of the Earth. A visual representation of this idea, can be achieved, if you draw an ellipse to represent a cross section of the earth. The angle that a normal to the ellipse makes, with the major axis of the ellipse, is the geodetic latitude. This angle varies non-constantly as the position moves along the ellipse around the Earth.

A third type of latitude can be designated from that same position on the Earth’s surface called the geocentric latitude. This is the angle that a line drawn from the point to the center of the Earth makes with the major axis of the ellipse.

And, a fourth type of latitude, is the reduced latitude, which is similar to the eccentric anomaly of a planetary orbit.

The Geoid is almost never coincident with the surface defined as the boundary between the solid and liquid parts of the Earth and the atmosphere. As Gauss said, it coincides in part with the surface of the oceans. On the solid parts of the Earth, the Geoid is the surface were the oceans extended under the continents in tiny channels. The Geoid is also different than the ellipsoid to which the geodetic, geocentric, and reduced latitudes refer. It was one of Gauss’ major breakthroughs to be able to map positions on the Geoid onto an ellipsoid. In this context, his general theory of curved surfaces emerged.

How can something which seemed to be so simple and stable, as the latitude of a position on the Earth’s surface, become so complex? After all, this is a real place on the hard surface of the Earth, isn’t it?

Well, that view is obviously a delusion. We cannot even think of a fixed place on the surface of the Earth, except in the context of the relationship of that place to the many astronomical and geodetic cycles. To even think of a fixed position on the surface of the Earth, we must take into our minds these many cycles. So the “true” surface of the Earth, is the interconnected surfaces which we conceive of in our minds, some of which we have just delineated. It is in this context, that the concept of Riemannian surfaces arises.

From this perspective, take a new look at Gauss’ first triangle, from Goettingen to Hohenbogen to Hils. The relationship among these three positions, is no longer a simple triangle, but must be considered as a projective relationship among these positions with respect to the “surfaces” of the Earth.

To determine this, first, we take the astronomical measurements at these three positions. So, one relationship, is the triangle formed in this way.

An entirely different triangle arises when we consider the positions along the surface of the Earth. Each position has a geodetic latitude and longitude, which, as we’ve seen, is different from the astronomical one. So we now have two different determinations of position of each point, the astronomical and geodetic.

The sides of this triangle are the shortest lines that connect them, called geodesics. On a plane, the geodesic is a straight line. On a sphere, the geodesic is a great circle. It is important to make the distinction between a great circle and a small circle on a sphere. The great circles are those circles whose centers coincide with the center of the sphere. All circles of longitude are great circles, but all circles of latitude are small circles. The significance of this distinction can be seen when considering two points on the same latitude. The circle of latitude that connects them is not the geodesic. But, a great circle can be drawn that connects the points, which is a shorter distance than the distance along the latitude.

So, in measuring the distances between Goettingen, Hohenbogen, and Hils, Gauss was measuring the length along the geodesics which connect them.

This intersected Gauss’ efforts to free mankind from the illusion of Kantianism. While it is obvious that triangles on the surface of the Earth are not Euclidean plane triangles, it is not Euclidean geometry that Gauss is trying to free us from. It is the Kantian idea, that any geometry must be taken as “true” a priori, that Gauss is overturning.

We have previously discussed, that the length along the sides of these triangles varies with the curvature of the sides. This seems to create the paradoxical situation, that to determine the length, we have to know the curvature, and to determine the curvature, we have to know the length.

Gauss’ conceptual breakthrough out of this paradox, was to consider, the triangle a function of the surface, and then determine the characteristics of the surface from the nature of the triangle.

In an letter to Gerling dated April 11, 1816, in which Gauss discusses Legendre’s theory of parallel lines, he ends saying:

“It is easy to prove, that if Euclid’s geometry is not true, there are no similar figures. The angles of an equal-sided triangle, vary according to the magnitude of the sides, which I do not at all find absurd. It is thus, that angles are a function of the sides and the sides are functions of the angles. naturally, such a function occurs, at the same time as a constant line. It appears something of a paradox, that a constant line could possibly exist, so to speak, a priori; but, I find in it nothing contradictory. It were even desirable, that Euclid’s Geometry were not true, because then we would have a priori a universal measurement, for example, one could use for a unit length, the side of a triangle, whose angle is 59 degrees, 59 minutes, 59.99999 seconds.”

Next week, we will look more at this relationship between triangles and surfaces, and between the triangle and a fourth point.

Surfaces, Triangles, and Projections

by Bruce Director

We begin with the section from Gauss’ letter to Gerling that we ended with last week:

“It is easy to prove, that if Euclid’s geometry is not true, there are no similar figures. The angles of an equal-sided triangle vary according to the magnitude of the sides, which I do not at all find absurd. It is thus, that angles are a function of the sides and the sides are functions of the angles, and at the same time, a constant line occurs naturally in such a function. It appears something of a paradox, that a constant line could possibly exist, so to speak, a priori; but, I find in it nothing contradictory. It were even desirable, that Euclid’s Geometry were not true, because then we would have a priori a universal measurement, for example, one could use for a unit length, the side of a triangle, whose angle is 59 degrees, 59 minutes, 59.99999 seconds.”

When Gauss wrote this in 1816, he was beginning his work on geodesy and was very much engaged in research in astronomy and terresterial magnetism. He would begin the great triangulation of Hannover several years later, where these considerations would play an important role. Over the last several weeks, we have been examining the first triangle in that survey, between Goettingen, Hohenbogen, and Hils. The ability to determine the lengths of the sides of this triangle, from the measurement of the angles, depends, obviously, on the principles of similar triangles, which as his letter to Gerling indicates, Gauss recognized was a function of the curvature of the surface.

Euclidean principles of similar triangles hold true, if the surface on which those triangles lie has no curvature, such as a plane. In such a case, two triangles can have exactly the same angles, but their sides can be of different lengths. The corresponding sides of the two triangles will all have the same proportion, but the absolute magnitudes will be different. To determine the exact size of a triangle on this surface, one needs therefore to know two angles and the length of the included side.

Such is not the case on a curved surface, such as a sphere. In this case, the angles of the triangle also determine the size. For example, think of the triangle formed on a globe (not the Earth) whose vertices are: the north pole, the intersection of the 0 degree meridian and the equator, and the intersection of the 90 degree meridian and the equator. This triangle, which contains 1/8 of the surface of the globe, has three 90 degree angles, for a total of 270 degrees. Compare this with the triangle formed when we change one of the vertices to the intersection of the 45 degree meridian with the equator. This triangle is half the area of the first, (1/16 of the surface of the Globe) and has two 90 degree angles and one 45 degree angle, for a total of 225 degrees. (The reader is encouraged to investigate this further by experimenting with other shapes and sizes of triangles. You can use a globe for this, or if you have a spherical object, such as a Lenart sphere, on which you can draw and measure.)

On both surfaces referenced above, the angles and sides are functions of each other, but the characteristics of these functions are entirely different on each surface. For example, the sum of the angles of a triangle, is not fixed a priori, but is determined by the curvature of the surface. On the plane this sum is always equal to 180 degrees, while on the sphere that sum varies with the size of the triangle.

Another crucial difference, is that on a surface of zero curvature, any triangle can be inscribed in a circle, and there is no limit to the size of that circle, hence no limit to the size of the area of the triangle. (The sum of the angles, of course, will always equal 180 degrees no matter how big the triangle.) A curved surface, however, is bounded. Thus, there is a maximum circle, and consequently, a maximum triangle. For example, on a sphere, great circles are the largest possible circles, and no triangle can be formed that is greater in area than the area contained in a great circle (half the area of the sphere).

In a letter to Taurnius dated November 8, 1824, Gauss discusses the relationship of the sum of the angles of a triangle, and the curvature of the surface on which the triangle lies. At the end of the letter, he reiterates the implications of the determination of a constant length for physical science: “… But it seems to me, we know little or nothing at all about the true nature of space, despite the actually meaningless rhetoric (Wort-Weisheit) of the Metaphysiker (I. Kant-bmd), than that we confuse something that occurs unnaturally to us with something that is absolutely impossible. Were the Not-Euclidean geometry the true one, and those constants lie in some relationship to such a magnitude, they can be determined a posteriori, in the domain of our measurements on the Earth and the heavens. That is why I have expressed the wish, occasionally in jest, that the Not-Euclidean geometry were not the true one, because then we would have an absolute measure a priori.”

In measuring the triangles in his geodetic survey, Gauss never wavered on his fundamental conviction that the nature of space cannot be determined a priori. As we’ve discussed previously, the determination of the distances on the Earth’s surface, depends on the curvature of the surface, which, in turn, is what Gauss was trying to determine.

Two fundamentally different approaches could be taken. Suppose, after measuring the angles of a triangle on the surface of the Earth, the sum of the angles is not 180 degrees, but nearly so. Is that difference due to an error of measurement, the curvature of the Earth’s surface, or both, and, if both, how much is due to each? One way to deal with this, is to assume that the triangles being measured on the surface of the Earth were small enough, that they could be considered as if they were on a plane. Then the principles of Euclidean geometry can be applied. A correction can then be made for the curvature of the Earth, once enough triangles were combined so as to encompass a greater part of the Earth’s surface. The amount of correction, can be determined only by making an a priori assumption for an approximate shape of the Earth. Such an approach would not require any consideration of the nature of space.

Bessel defended this approach in a letter to Gauss on February 10, 1829. (This letter is a response Gauss’s letter of January 27, 1829 in which Gauss expressed his geometrical views. We quoted Gauss letter in geodesy pedagogy #6):

“I would be very upset if you let the `shrieking of the Booetiers’ deter you from publishing your geometrical views. From what Lambert has said and what Schweikart has expressed verbally, it is clear to me, that our Geometry is incomplete, and a correction should preserve it, which becomes hypothetically insignificant, when the sum of the angles of a plane triangle equals 180 degrees. That were the true Geometry, the Euclidean is the practical, at least for figures on the Earth….”

To which Gauss replied on April 9, 1830:

“My true friend, it is easy for me to go into my views about geometry with you, especially since you are at least open to it. It is my conviction, that the nature of space has for our knowledge a priori, an entirely different position than the nature of pure magnitudes; there is for our knowledge, absolutely those things of whose necessity we are completely convinced (therefore, also of their absolute truth), of which the latter belongs; we must humbly admit, that if number is purely a product of our minds, space also has a reality outside of our mind, which we can not completely dictate…”

How did Gauss apply this in his geodetic survey? Instead of the “curve fitting” approach described above, and defended by Bessel, Gauss made no assumptions about the nature of the surface. How then, could he measure the triangles? Should he apply the Euclidean principles of plane triangles, the principles of spherical triangles, ellipsoidal triangles, or some other relationship?

Gauss approached the geodesy using the methods of modular and hypergeometric functions that we discovered in the Ceres series. As discussed over the last couple of weeks, the Earth is not one surface, but a multiply connected array of surfaces, each a reflection of some larger physical process. Measurement, therefore is determined, not with respect to one surface, but with respect to the projective relationship between these surfaces.

For example, we have shown how to determine the positions at Hohenbogen, Hils, and Goettingen, with respect to the astronomical cycles. The relationships among these positions with respect to those cycles is different than the relationship among the same positions with respect to the geodetic. In the latter case, each location as a corresponding set of co-ordinates on the surface of the Geoid (the surface that is everywhere perpendicular to the pull of gravity). So already there is a projective relationship between the astronomical relationships and the geodetic.

But, the Geoid is not the surface on which we are standing. The surface on which we are standing is either higher or lower than the Geoid. So we must be able to project the physical surface positions onto the Geoid and vice versa. Furthermore, the Geoid is not a regular surface, so to measure the relationship between Hohenbogen, Hils and Goettingen, we have to project the Geoid relationshipos onto a more regular surface, such as an ellipsoid. In other words, we are projecting positions on one irregular surfce (the physical) onto another irregular surface (the Geoid) onto a more regular surface (the ellipsoid) and continuing further onto a sphere and then onto a plane.

We must be able to make these projections, not just in one direction, that is from one surface to the next, but from any surface to any other surface.

By investigating what principles remain invariant under these projections, what principles change, and how they change, Gauss was able to determine a means to apply the principles of one surface to another. This required not only investigating the way one particular surface projects onto another, but the general principles of projection itself.

Investigations of projective geometry goes already back to the Greeks. The reader is probably familiar with the well-known stenographic projection of a sphere onto a plane. Here every point on the plane corresponds to a point on the surface of the sphere. Great circles on the sphere, project into lines on the plane, while small circles on the sphere project onto circles on the plane. We will look more deeply into this and other projections in future weeks.

Think also of Gauss’ discovery with respect to bi-quadratic residues. How do the relationships of prime numbers change when we extend the concept of number into the complex domain. (See pedagogy #97116bmd001.)

Gauss described this relationship between number and geometry in an April 1817 letter to Olbers:

“I become more and more convinced, that the necessity of our geometry can not be proven, at least not from human understanding for human understanding. Perhaps in another life we will come to another insight into the essence of space, which is now unreachable for us. Until then, one must not put geometry with arithmetic, purely a priori, but closer to in rank with mechanics …”

Complex Thinking

by Bruce Director

When we left off our last discussion on Gauss’ geodesy, we had discovered that the very concept of position, when considered in a physical sense — that is on this Earth — requires the mind to be able to think of more than one thing at once.

We must, for instance, consider the relationship of our position with respect to the motions of the Earth on its axis and around its orbit. We must consider our position with respect to the curvature of the surface of the Earth itself, which we have seen is not one surface, but an inter-connected array of surfaces. Additionally, the determination of the above considerations is a matter of measurement. That measurement, in turn, is dependent on the physical determination of the horizon.

In our example of the large triangle from Goettingen, the Meridian marker, and Hohenbogen, each location has an astronomical latitude and longitude, that mark the position as an angle between a heavenly body and the geoid (remember the geoid is that irregular surface that is everywhere perpendicular to the pull of gravity); a geodetic latitude and longitude that mark its position with respect to an ellipsoid that approximates the geoid; and a position on the physical surface of the Earth which may be higher or lower than the ellipsoid or geoid depending on the local elevation above sea level.

Here Gauss recognized a crucial paradox which he kept coming back to again and again. On the one hand, he insisted that, contrary to Kant, the true geometry of space, as he called it, could not be determined {a priori}, but was a matter of measurement. On the other hand, the principles on which that measurement was based are found only in the cognitive powers of the mind.

The same form of self-reflexive paradox was at the root of Gauss’ geodesy. For example, as we had earlier discussed, the ability to measure a distance along the surface of the Earth, depends on the curvature of the surface. But the only way to determine that curvature, is by measuring the distance.

If one faces this type of paradox, it is immediately evident that trying to divide the problem into components that dissolve the paradox fails. For example, in the triangle we have been examining, the components would be such simple magnitudes as, x degrees east, y degrees north, measured on the geoid, and x degrees east, y degrees north on the ellipsoid, and a “vertical” distance between the ground and the geoid and ellipsoid. One immediate problem that arises is that, since the ellipsoid is a regular surface and the geoid is irregular, the two can only coincide in one point. (The reader can illustrate this by putting an egg in a round bowl. Hold the egg still and notice that it touches the bowl in one place, while the rest of the egg is at varying distances from the bowl. Now tilt the egg so that it touches the bowl in a different point. All the previous relations have now changed.)

This Cartesian type of thinking is the equivalent of locating a position of a point in a plane as a horizontal and a vertical displacement with respect to an arbitrary origin. A Baby Boomer can fool oneself that such thinking appears to work, when, for example, marking off or measuring distances on a map, but from where does the map come?

This kind of paradox gave rise in Gauss’ mind to a revolutionary new concept that he identified as the complex domain. For Gauss, the complex domain was not a formal mathematical construct, but a required change in the mode of cognition, necessitated by a paradox in physical principles. Guass’ success in determining the orbit of Ceres, and his discoveries in geodesy, were demonstrations of the validity of this cognitive discovery.

Gauss recognized that the multiple relations among the positions in the triangle were not simple magnitudes which could be related to one another by an algebraic formula. Instead of imposing a linear grid on what is obviously a multiply-connected physical process, Gauss changed his thinking, according to the principles of classical metaphor, which, if you think about it, are totally congruent with the principles of well-tempered bel canto polyphony and classical poetry.

To illustrate the concept of the complex domain, Gauss gave a physical example we have cited before. Take a carpenter’s level and hold it in front of you. Under what rotations does the position of the bubble in the glass tube change? Once you get the level into a position so that the bubble is in the middle of the glass tube, hold the level at each end. The level can be rotated around three different perpendicular axes. Under which rotations does the bubble change its position?

Now extend the experiment. Place two levels at an angle to one another on a table that is free to move in three perpendicular directions. (This may be hard to obtain, but if you can get a hold of telescope or camera tripod that would be ideal. Otherwise, we urge the reader to be creative.) Now, adjust the heights of the legs of the tripod so that the bubble in one of the levels is in the middle of the glass tube. Unless you were extremely lucky, the bubble in the other level will not be in the middle of its tube. If you adjust the table to level the second level, you will disturb the first level.

In the first example, the line going through the glass tube of the level cannot be thought of simply as a line in a plane, but rather a line in a doubly-connected surface. The back and forth motion of the bubble in the tube is inseparable from the up and down motion of the ends of the level. Similarly in the second example, the surface of the table cannot be thought of simply as plane in space, but as a triply-connected surface.

And so, in Gauss’ mind, the only appropriate magnitudes to associate with this physical phenomena are complex numbers, or numbers that are connected to one another, not be simple linear relationships, but by a two-fold relationship.

I don’t mean to be abstract, but it is worth recalling here Gauss’ letter to Hansen: “These investigations lead deeply into many others, I would even say, into the metaphysics of the theory of space, and it is only with great difficulty that I can tear myself away from the results that spring from it as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”

To bring alive this concept of complex number, think back to the triangle between Goettingen, the meridian marker, and Hohenbogen. Instead of thinking of each vertex of the triangle as a combination of simple magnitudes, we can think of the geoid and the ellipsoid as two complex surfaces. The relationship between these two surfaces was at the heart of Gauss’ famous investigations into conformal mapping which will be explored in future weeks.

Gauss also investigated the complex nature of these surfaces by a study of a famous problem of geodesy which was known as the Snellius-Pothenot problem. The problem, originally posed by Snell and initially solved by French General Pothenot, is characteristic of the kinds of problems investigated by Carnot in his famous Geometry of Position.

The problem is this. In carrying out our triangulation, we have determined the positions of the vertices of one triangle. Now, we find a fourth point, either inside or outside that triangle, from which we can see the other three positions. How can we determine the position of the fourth point simply by measuring the angles of the lines of sight to the other three?

Another example of the same problem: suppose we are in a ship, and we can see three markers on the shore whose positions are marked on the ship’s chart. How can we determine our position by simply measuring the angles between the lines of sight from the ship to the three markers on the shore?

Next week, we will look at both Pothenot’s solution and Gauss application of the complex numbers to this problem.

Geometry of Position; A Concrete Example

by Bruce Director

The problem posed last time, to determine a position on the Earth, from the angles formed by the lines of sight from that position to three already-known positions, is a classic example of the “Geometry of Position.” The problem, known as the Snellius-Pothenot problem, occupied Gauss’ attention from his early student days into his years as a master geodesist. These investigations led him into a discovery of an application of his concept of the complex domain, demonstrating the congruence of this new idea with valid physical principles.

Working through this problem ourselves, affords us an opportunity to examine, in a concrete case, the subjective aspect of measurement, that is at the heart of Carnot’s Geometry of Position and Gauss’s physical geometry. This week, we will conduct certain experimental tests, designed to bring the crucial paradoxes into view. Next week, we will investigate the implications of these paradoxes.

The type of conditions that are represented in the Pothenot problem arise in a variety of circumstances. For example, a sailor needs to determine his position at sea, based on sightings of landmarks, such as lighthouses, buoys, or geographical landmarks whose positions are marked on the ship’s chart.

Or, as the problem presented itself to Gauss, and every geodesist from Snellius on: how to extend a triangulation measurement. For example, after Gauss had determined the exact positions of the observatory at Goettingen, the tower at Hohenbogen, and the Meridian marker, measured the angles between them, and accounted for the curvature of the Earth, he next chose a fourth position, from which the other three were visible. This fourth point X will form a new triangle attached to the first. He then had to determine the exact position of X. By adding triangles in this way, a large portion of the earth’s surface can be measured.

To situate the problem more concretely, the reader is encouraged to get out of your armchair and find a location from which you can see far off into the distance. Identify three distant objects and measure the angles between the lines-of-sight from your location to each of the objects. If you had already determined the latitude and longitude of these objects, how would you determine your current position and the distance from these objects, solely from the angles you just measured? This is the Pothenot problem.

Obviously, to make accurate measurements one would need a good theodolite or transit. However, you can improvise by cutting a circle out of wood or stiff cardboard. Mark on the circle diameters every 10 degrees. Place a nail or tack at the 0 degree mark on the circumference of the circle. Attach one end of a flat strip of wood (or cardboard) to the center of the circle so that the other end can pivot around. Place another nail on the free end of the strip. To measure an angle between two distant objects, hold the circle horizontally at eye-level, sight one of the objects with the nail at the 0 degree mark, and then pivot the movable strip until the nail on the end of it lines up with the second object.

Circumstances such as these require us to be able to determine something definite, i.e., a precise position, that is not susceptible to direct linear measurement. Instead, we must adduce the required position solely from the general characteristics of the physical domain. This limitation may at first appear to be a source of imprecision. But, in fact, this irony compels us to the type of cognitive action that enables our minds to generate entirely new conceptions concerning the nature of both the physical universe and our own minds.

As a first step toward gaining some insights into the geometrical relationship involved in the Pothenot problem, make a drawing representing the three known locations of Gauss’s first triangle. (For purposes of this illustration, make the triangle G-H-M a long, thin one, with one side G-M approximately 5 inches and angle GHM approximately 140 degrees.) Mark another point X, some distance away from triangle G-H-M.

Begin by examining the relationship between X and only two of the other known points, G and H. Draw straight lines connecting X to G and H. These lines represent the lines-of- sight. Using a protractor, measure the angle GXH. What other positions of X exist, whose lines-of-sight to G and H will form that same angle? Or, put another way, draw another line from H, in the general direction of X, but slightly different from HX. At what point on this new line will an observer see G and H at the same angle as GXH? To determine the manifold of all such positions, place the paper on some heavy cardboard and put two nails or tacks at points G and H. Now take two sticks (long bamboo barbecue skewers work well) and fasten them securely together so that they make the same angle as GXH. Place the sticks on the paper so that the ends of the sticks touch the nails and the vertex of the sticks is at X. If you slide the sticks along the nails, the position of the vertex will change, while the angle between the position of the vertex, and G and H, will remain constant. You should be able to now see that there are an infinite number of positions for X, from which an observer will measure the same angle with respect to the lines-of-sight to G and H. What is the curvature of the manifold of all such possible positions of X?

With some careful thought, you should be able to recognize that these possible positions of X, all lie on a circle that passes through G and H. This demonstrates that the exact position of X cannot be determined by its relationship to only two other positions.

Now look at the third point M, and connect it by a line to H. Connect X to H by a straight line that represents the line of sight from X to M. Take two other sticks and fasten them together so that they make the same angle with each other as the lines-of-sight HXM. Place a tack or nail at M. Now repeat the same process with respect to H and M, that you just carried out with respect to G and H. This will form another circle which is the manifold of all positions from which an observer at X will measure the same angle between the lines-of-sight to H and M.

Now we have constructed two circles. The first is the manifold of all positions from which an observer at X will measure a certain angle between the lines-of-sight to G and H. The second circle is the manifold of all positions from which an observer at X will measure a certain angle between the lines-of- sight with respect to H and M. The two circles will intersect at the position from which an observer at X will measure both angles simultaneously. This expresses the geometrical relationship of the position of X with respect to G-H-M. The actual calculation of the exact position of X, and its distance from G, H, and M, we leave for now, as an exercise for the reader, to be discussed in future weeks.

But that’s not all there is to it. If we now re-trace our steps, we can discover some crucial singularities, whose existence opens up a whole new domain of inquiry. You may have noticed when constructing the manifold of positions with respect to G and H, that as you changed the position of X, the distance from X to G increased while the distance from X to H decreased, or vice versa. A similar phenomenon occurred when constructing the circle with respect to M and H. The moment X reaches either G, H, or M, the continuous process of circular action is suddenly disrupted. The angle GXH ceases to be.

When we pass beyond this singularity, the angle GXH or HXM immediately changes its orientation. That is, depending on which side of the lines (GH or HM) X lies, the angles GXH or HXM change from acute to obtuse, or vice versa.

And there are other matters to be considered. Under what conditions is it not possible to determine a solution? Is it always possible to construct two distinct circles? What are the physical characteristics that were not taken into account in our geometrical approach? Investigating these matters led Gauss into what he called a “dainty” application of complex numbers, that we will investigate next week.

Number and the Geometry of Position

by Bruce Director

Hopefully, the reader took the advice given last week, and took the time to go to a location from which you could visually sight three distant distinct objects. This vantage point would have afforded you the opportunity to recognize the physical significance of the geometrical relationships between your position and the positions of the three distant objects. Such a relationship, may, as we tried to do last week, be representable on paper by drawings of geometrical shapes. But, these drawings must be considered, as with musical notes, only the footprints of the idea embodied in the musical composition. Yet, through these drawings and the geometrical relationships embedded in them, our minds, jumping o’er space, establish a connection between material objects far beyond our physical reach, as in a musical composition, where the opening motivic idea is united with the final cadence, simultaneously, even though they are separated in time.

With these thoughts in mind, turn back to the drawings we produced last week. Our position, X, whose position we do not know, was shown to lie at the intersection of two circles. Each circle represents the manifold of all positions that would form the same angle between the lines of sight from X to G and H, and from X to H and M.

It still remains for us to be able to determine the exact size of the circles, so that we can in turn determine the actual distance from X to the known positions G, H, and M. This is easily accomplished by some simple relationships of Euclidean geometry. You should have noticed that the line connecting G to H is a chord of the circle on which G, H and X lie. The perpendicular bi-sectors of any chord of a circle also intersect the center of the circle. Consequently, the center (which we’ll call O) of the circle G, H, X lies on the perpendicular bisector of G-H. But where? Another principle of circular action, is that the angle formed by the lines X-G and X-H, will be one-half the angle formed by the lines connecting the center of the circle to G and H, that is O-G and O-H, those lines being radii of the circle. Now we have a new triangle, O, G, H, that is a “projection” of the triangle X, G, H. Since O-G and O-H are radii, the triangle O, G, H, is an isosceles triangle, whose base angles H-G-O and G-H-O are equal. (I’m asserting a lot of relationships here without proof that are developed in Euclid’s Elements. These are undoubtedly subjects for future pedagogical discussions themselves, but for purposes of the present discussion, simply stating these principles is necessary for reasons of time and space.)

Using these relationships, it is easy to determine the location of the center of the circle. The base angles are equal to 180 degrees minus angle G-O-H divided by two. Using a protractor, these we can draw lines emanating from G and H at the calculated angle and the place where those lines intersect the perpendicular bi-sector of G-H, is the center of the circle on which G, H, and X lie. (This is a very simple construction to draw, even though it sounds complicated in words.)

A similar construction can be drawn with respect to the circle on which X, H and M lie. There is also a third relevant circle, that is the one whose center is the intersection of the perpendicular bi-sectors of both G-H and H-M. That is the circle on which the triangle G, H, and M lie. Keep these three circles in mind, as we will come back to them in future discussions.

There is the singular case, in which the two circles, (X, G, H and X, H, M) will coincide, if X, G, H, and M all were to lie on the same circle. In this singular case, it would not be possible to determine the position of X from the sightings to position G, H and M, and we would have to seek at least one different location before we could determine these positions.

This is a geometrical boundary condition, but what are the physical ones?

Gauss took up this question in a series of letters to Schumacher in April of 1836, as part of his relentless campaign to eradicate the influence of Kantianism from science.

For this, Gauss expanded on a principle developed by Carnot in his “Geometry of Position” that Schumacher had translated into German in 1810.

In {Geometry of Position}, Carnot polemicizes that a purely abstract conception of negative numbers is absurd. Negative numbers cannot be something less than nothing, as such a concept has no physical meaning. Instead, negative numbers, must be thought of only with respect to position.

For example, think of a triangle ABC. (Draw this triangle with vertex A on the top, B on the lower left and C on the lower right). On side BC, locate a point D. What is the length of the line segment BD minus BC? If point D lies between B and C, then this quantity will be negative. If D lies to the right of C then this quantity will be positive. The quantity is never less than 0, or less than nothing. Yet, it can still take on a negative value. Carnot suggested that negative and positive numbers be called lateral and reverse, not greater or less than nothing.

Whether this quantity is negative or positive, therefore, is not a function of its quantity, {but of its position}. Or, put another way, the sign of the quantity is a reflection of our hypothesis concerning the physical relationships of the positions A, B, C and D. As Gauss put it, in purely analytical procedures, one can always find a result. Whether that result corresponds to a physical reality, cannot be determined based on those numbers.

We already confronted one aspect of this problem last week when we discovered how the orientation of the angle, X, G, H changes from acute to obtuse at the singular points G and H.

In his April 1836 correspondence with Schumacher, Gauss stated, “The data of the Pothenot problem must necessarily contain:

“1) At least the mutual positions of the three given points (G, H, M in our example), must be completely determined, so that whether one lies to the right or left is unambiguous. It is therefore insufficient to know the magnitude of the angles, but also their position; therefore, for example, not merely A = 45 degrees, B = 45 degrees, C = 90 degrees, because these triangles are not different from one another. (Gauss draws two right isosceles triangles with opposite orientations. The difficulty expressing this without a picture is proof of the validity of Gauss refutation of Kant.)

“2) It is therefore, for example, insufficient to say that X makes an angle of 40 degrees between A and B and 100 degrees between B and C, because it would not represent one undifferentiated case, or in other words, because one would have not one problem, but at the same time four different problems.”

To illustrate this, the reader should draw a point X and then three lines from X to points A, B, and C so that C is above and to the right of X. Then draw B such that angle C, X, B is 100 degrees, and then draw A so that angle B, X, A is 40 degrees. B should lie between C and A. Now think of the different positions that A, B and C can take and still form the same angles with X. If you change one of the positions by 180 degrees, the same angular relationships will exist between the positions, but it will represent an entirely different physical problem, and in fact, may be physically impossible.

Just as Carnot identified that negative and positive numbers were better considered as lateral and reverse, Gauss recognized that the numbers associated with problems such as the Pothenot problem were complex. In his theory of bi-quadratic residues, Gauss called the elements of these complex numbers, lateral, inverse, and direct. Next week we will explore the meaning of complex numbers more thoroughly.

Truthful Numbers

by Bruce Director

In his writings on the Pothenot problem, Gauss took great delight, in pointing out, in a simple way, how almost everyone who had worked on this problem before, missed the most obvious point. Over the years, many mathematicians had developed increasingly elaborate methods for calculating the position of an unknown point X, from the angles of the lines of sight to three known positions, A, B, and C. The problem Gauss enjoyed pointing out, was that none of these mathematicians, could tell, from their calculations alone, whether their answer was physically possible.

This becomes evident when we review the singularities we’ve discovered over the last two weeks. To do this, make the following drawing:

On a piece of paper mark a point X. From X, draw three lines, XA, XB and XC, such that XA makes a 100-degree angle with XB and line XB makes a 40-degree angle with line XC. For now, make sure that line XB is between, line XA and line XC. The lengths of these lines is arbitrary. Now draw the two circles, XAB and XBC as discussed last week.

You should notice that the center of circle XAB is on the other side of the line AB from X, while the center of circle XBC is on the same side of line BC as X. This is a reflection of the singularity that results by the position where the chords, AB and BC, cut their respective circles. That is, each circle represents the manifold of all points at which an observer at X will see the points, AB, or BC respectively. But, depending on which side of the chord your on, the angle BXA, of BXC will either be acute of obtuse. Or, in other words, you could get exactly the same mathematical solution, for all angles that are complements of each other. (A complement of an angle is 180 degrees minus the angle. For example, the complement of 50 degrees is 130 degrees. The complement of 140 degrees, is 40 degrees.)

Thus, two completely different angles, will yield the same mathematical result. Gauss pointed out, as we demonstrated last week, that there are 4 possible configurations of these positions that exist for any combination of two angles. That is, in our example above, there are 4 possible configurations that will give the same mathematical solution for a position X that makes angles, of 100 degrees with A and B and 40 degrees with B and C. Furthermore, Gauss pointed out, if one of these mathematical solutions is the correct one, the other three will be wrong.

In an 1840 letter to Gerling, Gauss said, “The metaphysical basis of this appearance is that for the observed directions, one uses nothing other than the straight lines that make certain angles with one another, in which those lines can be extended indefinitely on both sides, while the progress of light happens in only one direction, therefore it is the case, that one must exclude the objects whose position is backward.”

In an letter to Schumacher on April 13, 1836, Gauss wrote, “The question is now, how can one express, in the simplest way, the conditions that are physically possible, by an equation between the data. I have often wondered, that this problem, about which so many have written, yet all of them, as far as I know, have overlooked entirely the essential circumstances — the solution I often discuss in my lectures….”

In a later letter to Schumacher, Gauss noted that in a concrete case, that is, one where you have actually done the measurements, it would seem that this problem would not arise, because you would already know the orientation of the angles. But, Gauss noted, it would be desirable to resolve this matter, because if there were a writing error, or some other mistake, it would be impossible to determine whether the result is correct from the calculations alone.

(Obviously, Gauss was interested in the more fundamental reasons for resolving this problem. But he clearly wanted to preempt any pragmatic arguments for not considering the metaphysics involved, by some smart aleck who thought such questions have no practical benefit.)

This is a similar problem to that identified by Carnot in the Geometry of Position. As in the example we discussed last week, two line segments could have the same length, but because of being in a different positions, the calculation of that length will have either a positive or negative value. But, how can a length, be less than 0, i.e., negative?

Obviously, the concept of number as merely a scalar quantity, is not truthful, as it leads to paradoxes. For example, when we multiply two negative numbers together, we get a positive number. Does that mean that if you multiply something that is less than nothing, by something that is less than nothing, you get something that is more than nothing? What is the basis for such a result, other than an established rule?

If numbers obey rules, that have no physical significance, are we to ignore numbers altogether? Gauss didn’t think so, as he fought to establish “citizenship” for truthful numbers, that he called complex numbers.

In the Geometry of Position, Carnot suggested replacing the idea of positive and negative numbers with the idea of lateral and reverse numbers. But, this doesn’t get rid of the paradoxes such as the one mentioned above about multiplying two negative numbers together.

Gauss used the problem of determining the physical criteria of the Pothenot problem to demonstrate the truthfulness of his conception of complex numbers.

Each of the positions in the Pothenot problem, can be defined by a direction and a distance from some other fixed position. The direction is measured by an angle, i.e., rotation, and the distance is measured by an extension. So, to Carnot’s idea of lateral and reverse numbers, Gauss adds a new dimension, calling them inverse numbers. The more familiar name for the combination of all three are complex numbers.

In other words, if we think of lateral numbers being in one direction, and reverse numbers be extension in one direction, inverse numbers are rotation. The units of these numbers are 1, -1 and the square root of -1, or i.

For operations on these numbers we no longer have to rely simply on rules.

For example, multiplication by a positive (lateral) or negative (reverse) number can be thought of as a 180-degree change in direction. In this way, if two reverse (i.e., negative) numbers are multiplied, the action is a 180-degree rotation, and the corresponding change in extension. Multiplication by an inverse number (i) is a 90-degree rotation.

Addition of complex numbers is identical to the process by which we determined the middle position of Ceres P2 by “adding” the segments O-Q1 and O-Q2 according to the “parallelogram” law.

(See Chapter 10, pp. 45-46 of the new Fidelio. For those who worked through this part of the Ceres problem, you should remember that the segments O-Q1 and O-Q2 were determined by the times elapsed between the observation of Ceres between its first and second position and its second and third position.)

If we associate a complex numbers with Q1 and Q2, their sum, will be the complex number associated with P2.

Now, back to the Pothenot problem. Gauss found the basis for determining the solution to this problem, if he were to think of each of the four positions as a complex number. By carrying out the indicated operations on these complex numbers, a unique solution could be found that would correspond to physical reality, and not suffer the falsity of scalar numbers. Thus, the complex numbers, as Gauss conceived them were more truthful numbers.

The best thing for the reader to do at this point, is some experimental work, using the tools developed by Gauss in his theory of bi-quadratic residues. Next week we will continue this experiment.

We append to the end of this discussion, a portion of the translation by Jonathan Tennenbaum, of Gauss’ announcement of his second paper on bi-quadratic residues. Using the visual representation of complex numbers that Gauss discusses, the reader should experiment with multiplying and adding these complex numbers to one another.

“We must add some general remarks. To locate the theory of biquadratic residues in the domain of the complex numbers might seem objectionable and unnatural to those unfamiliar with the nature of imaginary numbers and caught in false conceptions of the same; such people might be led to the opinion that our investigations are built on mere air, become doubtful, and distance themselves from our views. Nothing could be so groundless as such an opinion. Quite the opposite, the arithmetic of the complex numbers is most perfectly capable of visual representation, even though the author in his presentation has followed a purely arithmetic treatment; nevertheless he has provided sufficient indications for the independently thinking reader to elaborate such a representation, which enlivens the insight and is therefore highly to be recommended.

“Just as the absolute whole numbers can be represented as a series of equally spaced points on a line, in which the initial point stands for 0, the next in line for 1, and so forth; and just as the representation of the negative whole numbers requires only an unlimited extension of that series on the opposite side of the initial point; so we require for a representation of the complex whole numbers only one addition: namely, that the said series should be thought of as lying in an unbounded plane, and parallel with it on both sides an unlimited number of similar series spaced at equal intervals from each other should be imagined, so that we have before us a system of points rather than only a series, a system which can be ordered in two ways as series of series and which serves to divide the entire plane into identical squares.

“The neighboring point to 0 in the first row to the one side of the original series corresponds to the number {i,} and the neighboring point to 0 on the other side to {@msi,} and so forth. Using this mapping, it becomes possible to represent in visual terms the arithmetic operations on complex magnitudes, congruences, construction of a complete system of incongruent numbers for a given modulus, and so forth, in a completely satisfactory manner.

“In this way, also, the true metaphysics of the imaginary magnitudes is shown in a new, clear light….”

“Positive and negative numbers can be used only where the entity counted possesses an opposite, such that the unification of the two can be considered as equivalent to their dissolution. Judged precisely, this precondition is fulfilled only where {relations} between pairs of objects are the things counted, rather than substances (i.e., individually conceived objects). In this way we postulate that objects are ordered in some definite way into a series, for example A, B, C, D, … where the relation of A to B can be considered as identical to the relation of B to C and so forth. Here the concept of opposite consists of nothing else but {interchanging} the members of the relation, so that if the relation of (or transition from) A to B is taken as +1, then the relation of B to A must be represented by -1. Insofar as the series is unbounded in both directions, each real whole number represents the relation of an arbitrarily chosen member, taken as origin, to some determinate other member in the series.

“Suppose, however, the objects are of such a nature that they cannot be ordered in a single series, even if unbounded in both directions, but can only be ordered in a series of series, or in other words form a manifold of two dimensions; if the relation of one series to another or the transition from one series to another occurs in a similar manner as we earlier described for the transition from a member of one series to another member of the same series, then in order to measure the transition from one member of the system to another we shall require in addition to the already introduced units +1 and -1 two additional, opposite units +i and -i. Clearly, we must also postulate that the unit i always signifies the transition from a given member to a {determined} member of the immediately adjacent series. In this manner the system will be doubly ordered into a series of series.

“The mathematician always abstracts from the constitution of objects and the content of their relations. He is only concerned with counting and comparing these relations; in this sense he is entitled to extend the characteristic of similarity which he ascribes to the relations denoted by +1 and -1, to all four elements +1, -1, +i, -i.

“In form a concrete picture of these relationships it is necessary to construct a spatial representation, and the simplest case is, where no reason exists for ordering the symbols for the objects in any other way that in a quadratic array, to divide an unbounded plane into squares by two systems of parallel lines, and chose as symbols the intersection points of the lines. Every such point A has four neighbors, and if the relation of A to one of the neighboring points is denoted by +1, then the point corresponding to -1 is automatically determined, while we are free to choose either one of the remaining two neighboring points, {to the left} or {to the right}, as defining the relation to be denoted by +i. This distinction between right and left is, once one has arbitrarily chosen forwards and backwards {in} the plane, and upward and downward in relation to the two sides of the plane, {in and of itself} completely determined, even though we are able to communicate our concept of this distinction to other persons {only} by referring to actually existing material objects. [Kant already had made both of these remarks, but we cannot understand how this sharp-witted philosopher could have seen in the first remark a proof of his opinion, that space is only a form of our external perception, when in fact the second remark proves the opposite, namely that space must have a real meaning outside of our mode of perception.]

“But once we have made the second decision, we observe that it nevertheless depended on arbitrary choice, which of the two series intersecting [at A] we chose to regard as the primary series and which direction we assigned to the positive numbers; we furthermore realize, that if we instead take for +1 the relation originally denoted by +i, then we must necessarily denote by +i the relation formerly assigned to -1. In the language of the mathematician, however, this means +i is mean proportional between +1 and -1 or corresponds to the symbol \/-1; we have deliberately avoided calling i {the} mean proportional, because -i obviously has the same rights to that title. In this way we have completely justified a concrete conceptualization of \/-1, and nothing more is needed in order to admit this magnitude among the objects of Arithmetic.

“We hope to have done a service to the friends of mathematics with our brief presentation of the main points of a new theory of the so-called imaginary numbers. To the extent these numbers have heretofore been considered from an false point of view, and consequently have appeared as if surrounded by darkness and mystery, this is largely due to the unfortunate choice of adopted terminology. Had one called +1, -1, \/-1 direct, inverse and lateral units rather than positive, negative, imaginary (or even impossible), then there would have been no mystery at all. The author reserves the possibility of treating these matters, only barely touched upon in this paper, more fully at a later date, at which time we shall also answer the question, why such relations between things as form manifolds of more than two dimensions might not provide additional species of magnitudes to be admitted in general Arithmetic.”

Complex Polyphony

by Bruce Director

It is important to keep in mind, that the new numbers that Gauss had discovered, are themselves simply the “footprints” of a new metaphor, that more truthfully reflects the actual relations of space. Those relations, as we have seen in the various examples from geodesy, are not the simply-extended virtual reality of Kantianism, but actually, complex relationships, hence Gauss’ designation of them as complex numbers. To grasp this conception, the reader must not look to logically deductive mathematical references, but instead should seek the quality of thinking associated with classical artistic metaphor. Thus, Gauss’ conception, is completely distinguished, from the Cartesian, formal algebraic, obfuscation of complex numbers, that you may or may not have been taught in school. Regardless of your exposure to such classroom subjects, we all have some unlearning to do, before Gauss’ conception will come into view.

Gauss derived this new metaphor of complex numbers, from the interaction of the mind with the ordering of the physical universe. In the example we have been working through, the Pothenot problem, Gauss found a simple application of his conception of complex numbers, that demonstrated the greater truthfulness of this conception of space.

In previous weeks, we investigated the geometrical relationships among the four positions of the Pothenot problem. It was clearly demonstrated, that these relationships are a complex of angles and lengths, in which are embedded crucial singularities, that cannot be resolved by mathematical formalism.

(The reader will probably have trouble following the argument here, unless you actually performed some the experiments suggested over the last several weeks. Reports from those who have, confirm that the conceptions discussed, did not become real, until the person actually carried out the indicated task.)

Each of the known positions in the problem, A, B, C, are distinguished by a combination of a rotation (angle) and an extension (distance), either from the position from which we took the measurements, X, or from some other physical singularity, such as the north pole, or another known position on the Earth’s surface. In other words, the three known positions are related to each other, and the unknown position, by the doubly-connected action of rotation and extension. This combination of rotation and extension, references, in our minds, a complex number.

Now recall to mind, Gauss’s visual representation of complex numbers, as a quadratic array on a plane. This quadratic array can be thought of as a “map,” not of the positions themselves, but of the {relations} among the positions. On this “map,” Gauss can represent to the mind, the physical measurements of the Pothenot problem, as {relationships}, i.e., ideas.

(Again, the reader must think in terms of classical art for the proper references to grasp this point, not classroom mathematical logic. An appropriate mental reference is the well- tempered system of bel canto polyphony, whose relationships form an efficient means by which to communicate the process of creative discovery. The notes serve the thoughts. Complex mapping serves our thoughts concerning the relations of space.)

In a January 1821 letter to Olbers, Gauss discussed this method as generally applied to problems of geodesy. Out of this proposed application, Gauss developed his later theory of conformal mappings and curved surfaces. Riemann’s subsequent work on multiply-connected surfaces also flowed from the same fount.

The beginning of the letter discusses Gauss’ proposal for a geodetic survey of all of Europe:

“… My meaning is that after all well-measured triangles are in order, something should be considered, that is required for the public, that all of Europe should be covered by such triangles. For many years, I have drafted several methods, which would be suitable for such a measurement; though everything that I have read on the subject, I find thoroughly worthless. You have, for example, mathematicians, that have expended a great effort to the problem of calculating the latitude and longitude out of the distance from the meridian and the perpendicular, taking into consideration the elliptical form of the earth, while so far as I know, no one ever before questioned:

“1. How those distances, understood as one usually understands them, can be found from readings with greater accuracy; because it appears most of these calculations are done in a plane, or have been obtained by completely incorrect or useless methods;

“2. whether it were for the most part only suitable, to decide to use what is understood as distance, if one plans to derive it from inadequately precise triangles. This can only happen after the most arduous calculation, and from it, one can only with the greatest effort, descend to the latitude and longitude. All of this is like putting the cart before the horse. {Something usable should be put between the triangle and the longitude and latitude. It must be something entirely different than what is usually understood as coordinates.} How this appears in my theory, admittedly is too involved for me to explain here, I can only note as much, that what I put in between the triangle and the latitude and longitude, are those coordinates: 1) which are most suitable to be represented as points in a plane. Those coordinates follow most conveniently and easily from the measured triangle, and without assuming a very exact knowledge of the flattening of the earth; and 2) from it the latitude and longitude follow just as easily, in which one, naturally, must know the flattening. I have the intention, to make known this theory, no earlier than with my future measurements, and please for now, keep the application of these ideas to yourself. I will apply them not merely to my Hannover triangulation, but on all others that are linked to it, and so give a geometrical description of the greater part of Europe, if I can get the support for the undertaking.”

The coordinates Gauss suggested be interposed between the actual latitude and longitude, are the complex numbers, that suitably represent the relations among these positions. On this “map,” each of the positions is represented as a unique number, that is related to each other number in a doubly-connected way, as those positions are related to each other physically. That is, the relationship between the known positions, A, B, C, and the unknown position X, are related by the angles (rotation) formed by the lines of sight from X to A, B, and C, and the distances (extension) between them. Some of the distances we know, and some we are trying to determine.

Complex numbers, are related to each other in the same way. For example, if we take the position X as zero on our complex plane, and the latitude on which X lies as a line going through X, then the positions of A, B, and C are each determined by an angle of rotation from that line, and a distance from 0. The actual complex numbers that correspond to A, B, and C are still to be determined, but, as in all of Gauss’ work, it is the relations we are interested in, of which the actual numbers are simply a function.

So now, we have four complex numbers, X = 0, A = a + bi, B = c + di, C = e + fi. How can these numbers interact with each other?

One way, is that each of the numbers has a certain displacement from 0. We can add the displacement of one complex number to another. That is, as we indicated last week, by the “parallelogram law” that we employed in the Ceres orbit determination. If we draw the lines of sight from X, to A and B, these two lines can be represented in the complex domain as straight lines from 0 to a + bi and 0 to c + di. These two lines form two of the four sides of a parallelogram, and 0, a + bi, and c + di represent 3 vertices. We now complete the parallelogram, drawing a line starting at c + di that is parallel and with the same length as the line from 0 to a + bi. Similarly, draw a line from a + bi that is parallel and equal in length as the line from 0 to c + di. These two lines will intersect at the fourth vertex of the parallelogram. That vertex, call it D, will correspond to a new complex number, that is the sum of A + B, or (a + c) + (b + d)i.

We also indicated last week, that the action of multiplying by -1 is a rotation of 180 degrees and, multiplying by i is a rotation of 90 degrees. (Multiplying by 1 and -i is the same rotation, in the opposite direction.) Consequently, multiplying two complex numbers, say, (a + bi) x (c + di), involves combining the rotation and extensions of both numbers. In our example, the position of each number can be thought of as a rotation from, say, due east, and a distance from X. To multiply these two numbers, the rotations are added and the distances are multiplied.

Understood in this way, addition and multiplication of complex numbers are not simply rules of operations, but types of transformations of positions.

Now once again back to the Pothenot problem. Gauss, developed a new approach to a solution to the problem, by simple transformation of the angles of observation. This week we will investigate the problem geometrically. Next week, we will develop the application of complex numbers to the problem.

Let’s let Gauss describe the construction himself, as he did in a letter to Gerling written on Nov. 7, 1830:

“… About the fourth curious point in a triangle, I have never made known something myself. However, I informed Schumacher about it in 1808, and it appeared in the appendix of the second volume of his translation of Carnot’s {Geometry of Position} in 1810. I almost doubt that the point, which gave rise to the proposition that you cited, presents an otherwise especially elegant relation, and actually, is a one of those theorems that is only a special case of a more general one.

Particularly, if one constructs on the sides of a given triangle, A, B, C, three others, ABx, yBC, ACz, (either all three outside or all three inside triangle ABC), such that they are all similar to a second triangle, a, b, c, and, in such a way that the angles, x, y, z, are equal to a, b, c and are in a similar position; the lines Ax, By, Cz, (continued if necessary forward or backward), intersect at a point D and the angles at D (ADB, BDC, and CDA) are equal to the angles, x, y, z, or their complements, and are similar.

Obviously, this is the dainty construction of the Pothenot problem; but in praxis one can be satisfied to construct only one point, for example x, and the orientation on the surveyor’s table of the line Ax …”

Interlude: Complex numbers as musical intervals

by Jonathan Tennenbaum

Gauss’ approaches to the Pothenot problem and a whole series of other problems from astronomy, geodesy, and number theory, all pivot on his discovery of what is now called the {complex domain}. Complex numbers do not simply expand the repertoire of numbers already familiar to the high school student — whole numbers, negative numbers, fractions, irrational numbers, etc. With the complex numbers, Gauss introduces a {new principle} into mathematics, a principle that revolutionizes the very concept of number itself.

Most people today think of a {number} as the result of {counting or measuring something}. A complex number is neither of these, but rather signifies a certain kind of {relationship}. Complex numbers arise as a byproduct of analysis situs of a rather elementary sort. To get a sense of this, let’s amuse ourselves with a little dialogue.

Suppose there be given, anywhere on a plane, two similar triangles, i.e., triangles of the same shape but possibly different sizes and orientations. Take someone who is thinking in the mode which Lyn calls naive sense-certainty, and ask the person:

“What do you see here?”

“Well, two triangles,” comes the answer.

“Anything else?”

“No.”

“Are you sure there is nothing more?” Now the person leans over and scrutinizes the plane, glancing at us a few times with growing suspicion.

“No. There is absolutely nothing here except those two triangles.”

“And what about the {relationship} between the triangles?”

“A relationship? What is that? Show me where it is.”

“I mean, the triangles exist in the same Universe. So each one has a relationship to the other and everything else in the Universe — a relationship that would be different if either of the triangles were in a different location, or had a different size or shape, or were different in any other way.”

“Ah, now I know what you mean!” The person takes out a pencil, and connects each vertex of one triangle with each of the three vertices of the other triangle, creating a tangle of lines (otherwise known as “connectos”).

“No, no. A relationship is a different sort of thing than a visible figure, although we see everything we see only by virtue of relationships. What I am talking about is something like a musical interval.”

“You mean a pair of tones.”

“No. I mean the interval between them.”

“What? I hear nothing between them. Just two tones: now one, now the next one. Nothing more.”

“There you go again. Forget about listening to tones. Think about singing. If you sing one tone first, then in order to sing the next one you must {change} your pitch …”

“Oh. You mean the difference or ratio of the frequencies.”

“No. I mean the fact that a change occurs, of a specific sort. And we hear that change as an interval.”

“That’s funny. I only hear tones.”

“You haven’t understood. What I meant is, you hear the interval {in your mind}.”

“My {mind}? Where is that?”

At this point, we call upon a fellow organizer to assist the person in a very necessary search. Meanwhile, follow up the question posed by the dialogue: How might we conceive the relationship of two triangles, as a kind of {complex, geometrical interval} analogous to a {musical interval}?

Focussing on the {change} from one triangle to the other, notice that it comprehends three different aspects: First, as the triangles are located in different parts of the plane, there is an implied {change of overall position}. Second, the {size} of the triangles may be different — as for example, when the second triangle is twice the size of the first one — which means an implied {change of scale}. Third, even if the two triangles have the same size and shape, and occupy the same general position in the plane, they may have different {orientations}, which means an implied {rotation}. Thus, the {geometrical interval} between any two similar triangles involves {change in position}, {change of scale}, and {change of an angle} (i.e., rotation). Is it legitimate to conceptualize all three as constituting a single entity: a new type of number?

Gauss’ approach to this question was informed by the work of the Ecole Polytechnique of Gaspard Monge and the great General Lazare Carnot, whose later book “Geometrie de Position” marked an explicit return to the point of view of Leibniz. The immediate polemical focus of Carnot’s book was to demolish the banalized algebraic conception of numbers as merely {scalar magnitudes}. For this purpose, Carnot chose the long-debated issue of the existence or non-existence of {negative numbers}. We end this interlude with some quotes from Carnot’s famous book on the “Geometry of Position”:

“Precisely understood, Geometry of Position is the theory of so-called positive and negative quantities, or rather a means for eliminating that theory, which we completely reject … The greatest mathematicians could never agree on the meaning of isolated negative quantities [i.e., the idea of a quantity which is absolutely negative, as opposed to the expression of a {relationship} in the sense of analysis situs — JT]. And indeed, in order to obtain an isolated negative quantity, it would be necessary to substract a real quantity from zero; but, to remove a {something} from a {nothing} is an impossible act. How is one to imagine an isolated negative quantity, then?” “

Later Carnot reiterates again a whole series of paradoxes, connected with the naive notion of number as a mere counting or measurement:

“Some people regard negative quantities as smaller than zero; but it seems impossible to defend this view, since in order to obtain such a quantity it would be necessary to take something away from nothing. By the same token, -4 X -5 would have to be smaller than 2 X 3, since each of the factors in the first product would be smaller than the corresponding factor in the second. And yet, the first product yields +20, and the second only 6. Finally, given that in any calculation, all quantities equal to 0 can be disregarded, it would seem all the more justifiable to ignore quantities smaller than zero. But everyone knows what errors one would make, if one ignored negative quantities in a calculation.”

Carnot’s Geometry of Position, on the other hand, locates the reality of numbers not in simple magnitude per se, but in {changes} in geometrical relationships. I now quote Carnot’s own characterization. Although Carnot’s manner of expression will seem a bit obscure to most present-day readers, it is still worth quoting him, even without further explanation, to give a sense of the background of Gauss’ even more revolutionary conception:

“One must therefore reject all concepts of negative quantities as real existences … Everything amounts to replacing false and useless concepts by simple and true ones. I think I have succeeded in this, by replacing the controversial concepts of positive and negative, by the concept of what I call {direct} and {inverse} quantities.

“The quantities that I call direct and inverse, are ordinary, absolute quantities, each of which, however, is looked upon as being the variable difference between two other quantities, where one can sometimes become larger, sometimes smaller than the other. When the quantity which was larger at the beginning — i.e. in the (geometrical) configuration upon which the analysis was based –, {remains larger} than the other, then the quantity that represents the absolute difference of the two, will be called {direct}; when on the other hand the first quantity becomes {smaller} than the other, then the difference shall be called an {inverse} quantity. Thus the whole metaphysics of positive and negative quantities disappears. There remain only direct and inverse quantities, which are absolute and as good as any imaginable quantities. The signs put before these quantities in any given formula, must be retained or changed in accordance with the various circumstances in which they occur; and I call the theory of those changes Geometry of Position, because these changes express the different relationships of corresponding parts of figures of the same type.”

“The Geometry of Position investigates especially the connection between the respective positions of different parts of a given figure and their comparative values. There exist two different kinds of relationship between parts of a geometrical figure: relationship of magnitude and relationship of position. The first obtains between the absolute values of the quantities; while the other express their relative position, in so far as they show if a point is above or below a line, to the right or left of a plane, inside or outside a closed curve. My goal is to compare these two aspects of the relationship of geometrical quantities, and to bring them closer to each other.”

Playing With Numbers

by Bruce Director

If you worked through the last two pedagogical discussions, (Truthful Numbers 98266bmd001 and Complex Polyphony 98276bmd001) and the Interlude (98286jbt010), you should be beginning to get a mental glimpse of the metaphor of the complex domain, as developed successively by Gauss and Riemann, out of the discoveries of Leibniz and Carnot. This metaphor seeks to free the mind from the bondage of sense-certainty’s linearity, and present to reason the complex relationships of space-time.

But why express this in numbers, complex or otherwise? Aren’t these numbers simply formal representations? These questions reflect the illiteracy of our day. Cusa in “On Conjectures” writes:

“The natural sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality. So much, indeed, is number shown to be the beginning of those things which are attained by rationality, that with its sublation, nothing remains at all, as is proven by rationality. And if rationality unfolds number and employs it in constituting conjectures, that is not other than if rationality employs itself and forms everything in its highest natural similitude, just as God, as infinite mind, in His coeternal Word imparts being to things. There cannot be anything prior to number, for everything other affirms that it necessarily existed from it….

“The essence of number is therefore the prime exemplar of the mind. For indeed, one finds impressed in it from the first trinity or the unitrinity, contracted in plurality. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

Thought of in this way, the complex domain is nothing other than an unfolding polyphony. And the best way to discover its beauty, is, as Schiller notes, to play with it. Which is exactly what Gauss did.

When he was 17 years old, Gauss discovered, that the circle could be “knowably” divided into 17 parts, something believed to be impossible for over 2000 years. His mind unfolded this discovery, in the way Cusa says, through number. (To be discussed in a future pedagogical.) This concept of number is consistent with the crucial paradoxes investigated by Pythagoras and Plato that gave rise to the discovery of the “incommensurable” numbers. Gauss called these investigations of number, “higher arithmetic”. Speaking of these investigations, Gauss said he got more pleasure from playing in the domain of higher arithmetic than all his astronomical discoveries.

Upon studying Lazare Carnot’s “Geometry of Position,” (sections of which you were introduced to in the “Interlude”), Gauss enjoyed applying his new conception of complex number to the questions raised by Carnot.

One such application of complex numbers was written on the inside back cover of Gauss’ copy of Schumacher’s 1810 translation of Carnot’s “Geometry of Position,” titled “Construction of the Pothenot Problem.” Before developing that example, some preparatory play would be helpful. Of course, this play, requires some work, otherwise it wouldn’t be fun.

To begin, you must be in an elevated state, so first recite a poem, sing a song, or listen to some music. One cannot play in this domain in a rage.

Now, think about the relationships we’ve been investigating in the Pothenot problem. Three positions on the surface of the Earth have been determined with respect to astronomical observations. (This gives us, of course, a relationship among the three positions, not three different and separate sets of coordinates.) We then physically observe those three positions form a fourth position, and we now seek to determine the relationship of this fourth position, to the other three, and the astronomical cycles.

The relationship among these four positions is characterized by a change in the measured angle between the lines of sight, and the unmeasured distance, between the positions. These relationships are reflected by a complex number. So, Gauss represented these positions, in his mind, as complex numbers.

Here the true beauty of the complex domain, begins to show itself. Not in the relationship between points, but in the transformation of these relationships as a whole.

For example, think of a line segment. Or, better yet, think of the line as the “footprint” of an interval between two complex numbers. Now, multiply both complex numbers by i, which as we earlier discovered, means a rotation of 90 degrees. The interval between these two numbers, is also rotated 90 degrees. Now, multiply the new numbers by i, rotating them and the interval, another 90 degrees. Now, multiply these new numbers again by i, rotating them and the interval another 90 degrees, and then again which brings the line (interval) back to its original orientation and position. It should be easy to see, that the corresponding endpoints of these four lines, form the vertices of a square, and also lie on a two circles respectively.

(It is important that you try this with a concrete example. I’ve avoided giving one, as I don’t want to spoil your fun.)

Do the same thing, but instead of multiplying by i, multiply the endpoints by 1+i. Compare these two types of transformations.

Now, form a triangle between three complex numbers. Multiply the vertices of this triangle by i. Also try multiplication by other numbers, such as 3, -3, 1+i. How does this transform the triangle?

Try multiplying a complex number by itself, again and again.

If you play in this way, you can begin to see the types of transformations, that occur in the complex domain.

In the announcement to his second treatise on bi-quadratic residues, Gauss says, “Positive and negative numbers can be used only where the entity counted possesses an opposite, such that the unification of the two can be considered as equivalent to their dissolution. Judged precisely, this precondition is fulfilled only where <relations> between pairs of objects are the things counted, rather than substances (i.e. individually conceived objects). In this way we postulate that objects are ordered in some definite way into a series, for example A, B, C, D, … where the relation of A to B can be considered as identical to the relation of B to C and so forth. Here the concept of opposite consists of nothing else but <interchanging> the members of the relation, so that if the relation of (or transition from) A to B is taken as +1, then the relation of B to A must be represented by -1. Insofar as the series is unbounded in both directions, each real whole number represents the relation of an arbitrarily chosen member, taken as origin, to some determinate other member in the series.”

Under this concept, positive and negative numbers have no self-evident existence, but come in pairs that bound an interval of a line. Similarly, any complex number has an opposite “such that the unification of the two can be considered by their dissolution.” This defines a certain unique type of interval with respect to a complex number, such that (a+bi)-(c+di)=0.

Play with some specific examples to grasp the principle here.

This leads directly to a crucial relationship in Gauss’ application of complex numbers to the Pothenot problem, that is the relationship of three complex numbers, such that the unification of all three is equivalent to their dissolution. Geometrically this triple interval forms a triangle whose vertices are complex numbers that all add up to 0.

To construct such a relationship, begin with two complex numbers, (a+bi) and (c+di) that in turn define a unique interval. Add these two numbers, using the parallelogram principle, defining a new complex number (e+fi). Then determine the opposite of that new complex number (-e-fi). This number (-e-fi) will then form a triangle with (a+bi) and (c+di) that when all three are united, in any order, will equal zero.

This unique type of triple interval between three complex numbers is the key to Gauss’ solution of the Pothenot problem. But, enough play for this week.

Civil Rights for Complex Numbers

by Bruce Director

“It is this and nothing other, that for the true establishment of a theory of bi-quadratic residues, the field of higher arithmetic, that otherwise extends only to the real numbers, will be enlarged also to the imaginary, and these must be granted complete and equal civil rights, with the real. As soon as one considers this, these theories appear in an entirely new light, and the results attain a highly surprising simplicity.”

— Carl F. Gauss. Announcement to his second treatise on bi-quadratic residues.

If you conducted the experiments with complex numbers suggested in last week’s pedagogical discussion, you probably experienced a certain uneasiness, as the familiar principles of addition, subtraction, and multiplication took on strange new properties. That uneasiness did not arise because you were learning something new, but because you were unlearning something you didn’t even think you knew.

Just as Plato’s reflection on the discoveries of Pythagoras, unveiled the deeper implications of the paradoxes associated with the existence of “irrational numbers,” Gauss’ reflections on the paradoxes arising from the investigations of Cusa, Kepler, Leibniz, and Carnot unveiled the deeper implications of the principles underlying what he identified as the complex domain. The principles are not to be found in the formal representation, but, in the beauty that results when the form is united to the ideas.

A new look at the familiar actions of addition, subtraction, multiplication, and division of numbers, reveals underlying assumptions of which you were completely unaware, even though you were very emotionally attached to them. (We are excluding here the poor Generation X’er or Baby Boomer, who thinks that these arithmetic operations are just different buttons on their calculators.)

For example, you may have been perplexed by the discovery, that in the complex domain, addition and multiplication of complex numbers, seemed to be based on completely different principles, while with “real” numbers, addition and multiplication seem so similar.

Euclid in Book 7 of the Elements provides the following definitions:

“1. A unit is that by virtue of which each of the things that exist is called one.

“2. A number is a multitude composed of units.

“15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

“16. And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.”

From these definitions, it would appear that multiplication is simply the repeated addition you were taught in school. But, embedded in these very definitions, are underlying assumptions, the implications of which Plato reflected on in the Meno and Theatetus dialogues.

Embedded, but unstated, in these definitions of Euclid, is that this concept of number is associated with physical extension, and, as Plato points out in the Theatetus, the type of extension determines the type of number.

Thus, if we associate a number with linear extension, then multiplication seems to be simply repeated addition. But, if, as Euclid indicates in definition 16, multiplication changes the type of extension, from linear to planar, something new has been introduced — rotation. That, in turn, generates a new type of number, an “incommensurable” that cannot be generated in the domain of simple linear extension.

Additionally, as Carnot pointed out, there is another assumption underlying the concept of number — direction, giving rise to the true, i.e., non-formal, meaning of negative and positive numbers. Gauss, as we’ve seen, insisted that these characteristics, (direction and extension) cannot be determined formally, but only with respect to physical phenomena.

There are, of course, other obvious paradoxes, such as prime numbers, that reveal other relevant underlying assumptions about number, that we have touched on in previous discussions, and will return to in future ones.

(A musical analogy may be helpful here. Think of the principle of inversion of musical intervals, but, limited to only one key. Now think of that same principle of inversion, but in the domain of complete well-tempered system.)

In his application of complex numbers to the Pothenot problem, Gauss was not seeking his desired civil rights for complex numbers, by demonstrating their practicality. Gauss sought the beauty in the problem, which emerged when he confronted the impracticality, of seeking a solution through practical means.

From this standpoint, Gauss defined the once familiar principles of addition and multiplication, this way, in his second “Treatise on Bi-Quadratic Residues” Compare this with Euclid:

“… In this way it can be said, that those arbitrary complex magnitudes, measure the difference between positions of the points to which they belong, and the beginning point; if the positive unit describes an arbitrary, but definite deviation from an arbitrary but definite direction, the negative unit describes a just as great deviation from the opposite direction, and finally, the imaginary units a just as great deviation from the two perpendiculars on both sides of these directions.

“In this way, the metaphysics of magnitudes, which we call imaginary, will stand in the most excellent light. If the beginning point is described by (0), and both complex magnitudes m, m’ express the positions of points M, M’, with respect to the point (0), so the difference m-m’, will be nothing other than the position of point M with respect to M’; on the other side, if the product mm’ represents the position of a point N with respect to (0), it can be easily seen, that position will be determined by the position of M with respect to (0), as the position of M’ is determined by the position of those points that correspond to the positive unit, so that it would not be incorrect to say, that the position corresponding to the complex magnitudes mm’, m, m, 1, form a proportion….”

From this standpoint, having conducted some experiments, look back at the actions, addition and multiplication of complex magnitudes. Each such magnitude has a relation to some arbitrary beginning point identified with (0).

(While such a beginning point is formally arbitrary, it is definitely determined with respect to real physical processes. For example, on an abstract sphere, any arbitrary point can be chosen as a pole, which in turn determines an antipole and an equator. But on the Earth, the north and south pole and equator are determined by the physical process of the Earth’s rotation on its axis. In the Pothenot problem, the beginning point, is determined by the point X from which you are observing the known points A, B, and C and measuring the angles between the lines of sight XA, XB, XC.)

Before making the final step to Gauss’ application of complex numbers to the Pothenot problem, we should conduct one final preparatory experiment.

Consider three complex numbers, a, b, c, corresponding to three positions, A, B, C. According to Gauss’ conception cited above, each of these positions is related to each other AND some beginning point (0).

To grasp this two-fold relationship, form a triangle by drawing the lines AB, BC, and CA. These lines (length and direction) can be thought of as denoting the interval between the complex numbers, a, b, c. What is the relationship among these intervals, with respect to the beginning point 0?

Use Gauss’ conception, the relationship between the position of A and B, for example, is the difference of the corresponding complex numbers a-b. One determines that difference from the parallelogram formed by applying the displacement 0-b, to a, in the opposite (negative) direction of 0-a. This determines a new complex number, d that is equal to a-b. The complex number d corresponds to a new position D. D will thus have the same relationship to 0 as A does to B, and the line 0-D will have the same length and direction as AB.

Consequently, subtracting the complex numbers a-b, has the effect of transposing (modulating), the interval a-b, from the positions A and B, to the position corresponding to 0. Similarly, with the other intervals in the triangle ABC.

If you now carry out these three actions, a-b, b-a, c-a, you will have performed transposition (modulation) of all three intervals to the beginning point 0. This creates an inversion of the intervals associated with triangle ABC. Like a musical inversion, something very interesting happens, the implications of which lead directly to the solution to the Pothenot problem.

Gauss’ “Zierlichste” Construction of the Pothenot Problem; A Real World Solution

by Bruce Director

We now have all the elements in place to complete the construction of what Gauss called the “zierlichste” (“most elegant”) solution to the Pothenot problem. These elements are:

1) The intersection of the Earth’s motion with certain astronomical cycles, was measured at three different positions on the Earth’s surface. These measurements determine a unique relationship between each position, the astronomical cycles and the Earth’s motion;

2) The astronomical intersections of each position also interact with the physical geometry of the Earth itself, determining a geodetical relationship. This interaction is expressed as a relationship of angles and length, physically measured between the positions.

3) The interaction of the astronomical and geodetical relationships exist in a multi-dimensional multiply-connected domain, and cannot be expressed mathematically by a formal, i.e., Kantian notion of empty Euclidean space, but only by Gauss’ metaphor of the complex domain. Or conversely, the axiomatic assumptions of Kantian empty space, cannot be imposed truthfully on these physical relationships.

4) The Pothenot problem is an elementary problem of analysis situs. Determine the unknown interaction, both astronomical and geodetical, of a fourth position, from the observed relationship of that fourth position to the other three, already measured positions.

(Point 3 cannot be adequately grasped passively. On Wednesday, the participants of the weekly Gauss discussions in Leesburg discovered this for themselves, by measuring a physical angle with a surveyor’s transit. This involved levelling the instrument, taking the sightings, and reading the angle. To the surprise of many, there was a vast difference between taking that measurement, and measuring the angles between two lines on a piece of paper. Lyn’s repeated references, to the effects on the mind of a non-productive society, are very relevant to this point.)

Gauss’ solution to the Pothenot problem, provides us with a simple, but devastating, pedagogy to demonstrate that the formal notions of Kantian space are an illusion, and could not truthfully represent physical processes.

Extending the work of Cusa, Kepler, Leibniz, Monge, and Carnot, Gauss created the metaphor now known as the complex domain. Continuing the section of his second treatise on bi- quadratic residues, that we quoted from last week, Gauss states:

“Thus, we reserve for ourselves a more detailed treatment of these subjects for another opportunity. The difficulty, one has believed, that surrounds the theory of imaginary magnitudes, is based in large part to that not-so-appropriate designation (it has even had the discordant name `impossible magnitude’ imposed on it). Had one started from the idea to present a manifold of two dimensions (which presents the conception of space with greater clarity), the positive magnitudes would have been called direct, the negative inverse, and the imaginary lateral, so there would be simplicity instead of confusion, clarity instead of darkness.”

As you should have discovered by working through the experiments discussed over the last several weeks, numbers in the complex domain are intervals, and a relationship between two or more numbers, is a relationship of intervals to intervals. In the case of the Pothenot problem, it is the intervallic relationship of a triangle, that concern us.

Take the case of the triangle that we investigated at the end of last week’s discussion. The vertices of this triangle can each be represented by a complex number, such that those vertices are related to each other, AND, with respect to some beginning point, that is designated by the complex number 0. This relationship expresses itself, in the differences between the complex numbers that correspond to the vertices.

Let’s take a concrete example: A triangle whose vertices are represented by the complex numbers a=1+2i, b=3+2i, c=1+4i. (You must draw this yourself.) This is an isosceles triangle, with a 90-degree angle at a, and 45-degree angles at b and c. Now, take the differences between the vertices: a-b = -2; b-c = 2-2i; c-a = 2i; Call these intervals, p, q, r, respectively. These intervals correspond to three new positions. The relationship between p, q, and r is an “inversion” of the original triangle a, b, c. In the inversion, the line 0-(2i) forms a 90-degree angle with the line 0-(-2), just as the side a-b forms a 90 degree angle with side b-c, except, that the angle has the opposite orientation. Similarly, the line 0-(2-2I), is the same length and direction as side b-c, but forms a 135-degree angle to the other two sides.

It is of extreme importance, that we took the intervals all in the same direction. Now take the intervals in the opposite direction: that is, b-a = 2; a-c = -2i; c-b = -2+2i. Call these intervals, -p, -q, -r respectively. This inverts the triangle in a similar way, except in an opposite orientation.

In the first case, we took the intervals all in a counter- clockwise direction, in the second case, a clock-wise direction. Because all the intervals were taken in the same direction, the resulting positions have the important characteristic of adding up to 0. That is, the complex numbers, p+q+r=0 and -p+-q+-r=0.

Call these two inversions, inversions 1 and 2. Borrowing a musical analogy, I call these two inversions, “perfect inversions.”

Now, let’s mix the directions. For example, a-b = -2; c-b = -2+2i; c-a = 2i; This forms a new configuration, in which the line 0-(2i) and 0-(-2) form right angles, and the line 0-(-2+2i) forms a 45-degree angle between the other two.

There are six possible mixed inversions. Keeping to the musical analogy, call these inversions “imperfect inversions.” Each of the “imperfect inversions” is distinguished from the other, by a rotation of one of the intervals by 180 degrees.

Each of the “imperfect inversions” represents a possible configuration of the angular relationship between an observer and three other positions. If the observer were standing at 0, four of the inversions would represent configurations in which that observer measured a 45-degree and a 90-degree angle between the lines of sight to the three other positions. Two of the inversions, represent a configuration in which an observer would measure two 45-degree angles between the lines of sight to the three other positions.

From the standpoint of abstract formalism, it is impossible to distinguish these configurations from one another. For example, the four inversions that form one 45-degree and one 90-degre angle, are all mathematically possible solutions. Gauss showed that if one of them was the physically correct one, the other three would be physically impossible.

In an 1840 letter to Gerling, Gauss said, “The metaphysical basis of this appearance is that for the observed directions, one uses nothing other than the straight lines that make certain angles with one another, in which those lines can be extended indefinitely on both sides, while the progress of light happens in only one direction, therefore it is the case, that one must exclude the objects whose position is backward.”

These “inversions” therefore, express a characteristic about a triangle, that is not visible in the mere shape of the triangle.

Now, what happens when we perform the same “inversions” with a triangle that is similar, i.e., has the same angles as the original triangle, but different length sides? For example, a triangle whose vertices are represented by the complex numbers d=7+5i; e=11+5i; f=7+9i. This triangle has the same angles, as triangle a, b, c and the same orientation, but its sides are twice as long. Similarly, when the corresponding “inversions” are performed (taking the intervals between the intervals), similar configurations to the inversion of triangle a, b, c result.

A new insight into the meaning of similar triangles arises under this principle of inversion.

For example, if we take what I called a “perfect inversion” of triangle a, b, c, then the corresponding intervals, p, q, r, expressed as complex numbers, add up to 0. Each of these intervals, corresponds to a side of the original triangle a, b, c. Each side of the original triangle corresponds to a side on triangle d, e, f. That is interval p corresponds to sides d-e and a-b; interval q corresponds to sides d-f and a-c; and interval r corresponds to sides f-e and b-c;

Now, since the intervals p+q+r=0, how is this relationship expressed in any similar triangle? A beautiful principle emerges, such that if the complex numbers p, q, r are multiplied by the complex number corresponding to the opposite vertex, the result will also add up to 0. In our example, (p x f) + (q x d) + (r x e) = 0.

We see that similar triangles are related to each other by a characteristic of the inversion. That characteristic functions as a type of modulus. In the complex domain, similar triangles are not just arbitrarily floating around in space, but have a modular relationship with respect to the inversion.

This modular relationship, forms the basis for what Gauss’ called a “zierlich” solution of Pothenot problem in the complex domain.

The geometrical basis for the solution was discussed in pedagogy titled Complex Polyphony (98276bmd001) and is defined as follows: (Again, you must draw this.)

From a position designated X, we measure the angles between the lines of sight to three already determined positions A, B, C. Thus, a triangle is formed between the positions A, B, C and X is either inside that triangle, or outside it. The relationship of X to positions A, B, C, can be determined by forming an “auxiliary” triangle, x,y,z such that the angles of that triangle are equal to the angles measured between the lines of sight, X-A, X-B and X-C. (Or the complements of those angles if the angles are obtuse.)

Now construct on the sides of triangle ABC, similar triangles to x,y,z. (These triangles should be either all on the outside, or all on the inside of triangle ABC, depending on whether X is inside or outside triangle ABC.) These new triangles will have vertices A,B,x1; B,C,y1; and C,A,z1. If we then connect x1 to the opposite vertex of triangle ABC, that is vertex C; y1 to vertex A and z1 to vertex B, these three lines will intersect at one point, and that will be the position of X. In other words, X will be the position, at which an observer would have to be, in order to measure the angles between the lines of sight to positions, A,B,C.

The solution becomes very “dainty” in deed, if we designate these positions by complex numbers. In that case, we designate the three known positions, A, B, C by complex numbers. Then, we form the perfect inversion of the “auxiliary” triangle x,y,z. The inverted intervals, call them l, m, n, will add up to 0. To construct the similar triangles A,B,x1; A,C,y1;, C,A,z1; we need only multiply the complex numbers A,B,x1 by the corresponding intervals, l, m, n. That is, if A is opposite l, and B is opposite m, then x1 is that complex number, that when multiplied by n, forms the inversion (A x l) + (B x m) + (x1 x n) = 0.

Similarly, we can find the complex number that corresponds to y1 and z1 and from these numbers, determine the complex number that corresponds to X.

(Again, we apologize for the verbosity of this description, owing to the lack of use of diagrams in this means of communication. However, if you work slowly through the construction, it is not as complicated as these words make it appear. Also, for reasons of time and space, we leave the complete demonstration of this to a future pedagogical, or for the reader to work out on his own.)

This is just the beginning of Gauss’ application of complex numbers to problems of physical geometry. In future weeks we will investigate these more fully, including Gauss’ work on the pentagrama mirificum, conformal mapping, and his theory of curved surfaces. All this opens the door to Riemann’s extension of Gauss work on multiply-extended manifolds.

## The Epinomis and the Complex Domain: A Fragmentary Dialogue in the Simultaneity of Eternity

by Bruce Director

The following is provided to provoke some thinking, with respect to matters raised in previous pedagogical discussions, and to lay a conceptual basis for subjects to be taken up in the near future.

Plato’s dialogue of the Laws, continues in the short appendix known as the Epinomis:

“Let us then first consider what single science there is, of all those we have, such that were it removed from mankind, or had it never made its appearance, man would become the most thoughtless and foolish of creatures. Now the answer to this question, at least, is not overhard to find. For, if we, so to say, take one science with another, ’tis that which has given our kind the knowledge of number, that would affect us thus, and I believe, I may say that ’tis not so much our luck as a god who preserves us by his gift of it….

“But we must still go forth a little on our argument, and recall our very just observation, that if number were banished from mankind, we could never become wise at all. For a creature’s soul could surely never attain full virtue, if the creature were without rational discourse, and a creature that could not recognize two and three, odd and even, but was utterly unacquainted with number, could give no rational account of things, whereof, it had sensations and memories only, though there is nothing to keep it out of the rest of virtue, valor, and sobriety. But without true discourse, a man will never become wise, and if he has not wisdom, the chiefest constituent of full virtue, he can never become perfectly good, and, therefore, not happy. Thus there is every necessity for number as a foundation, though to explain why this is necessary, would demand a discourse still longer than what has gone before. But, we shall be right, if we say of the work of all the other arts which we recently enumerated, when we permitted their existence, that nothing of it all is left, all is utterly evacuated, if the art of number is destroyed.

“Perhaps when a man considers the arts, he may fancy that mankind need number only for minor purposes — though the part it plays even in them is considerable. But could he see the divine and the mortal in the world process — a vision from which he will learn both the fear of God and the true nature of number — ….

“Well then,… How do we learn to count?… There are many creatures whose native equipment does not so much as extend to the capacity to learn from our Father above how to count. But in our own case, God, in the first place, constructed us with this faculty of understanding what is shown us, and then showed us the scene he still continues to show. And in all this scene, if we take one thing with another, what fairer spectacle is there for a man, than the face of day from which he can then pass, still retaining his power of vision, to the view of night, where all will appear so different? Now as Uranus never ceases rolling all these objects round, day after day, and night after night, neither does he ever cease teaching men the lore of one and two, until even the dullest scholar has sufficiently learned the lesson of counting. For any of us who sees this show will form the notion of three, four and many. And among these bodies of God’s fashioning, there is one, the moon, which goes its way, now waxing, now waning, as it lights up one day after another, until it has fulfilled fifteen days and nights, and they, if one will treat its whole orbit as a unity, constitute a period, such that the very slowest creature, if I may say so, on which God has bestowed the capacity to learn, may learn it…. [W]hen God made the moon in the sky, waxing and waning, as we have said, he combined the months into a year and so all the creatures, by a happy providence, began to have a general insight into the relations of number with number. `Tis thus that earth conceives and yields her harvest so that food is provided for all creatures, if winds and rains are neither unseasonable nor excessive; but if anything goes amiss in the matter, ’tis not deity we should charge with the fault, but humanity, who have not ordered their life aright….

“… So we must do what we can to enumerate the subjects to be studied, and explain their nature and the methods to be employed, to the best of the abilities of myself who am to speak and you who are to listen — to say, in fact, how a man should learn piety, and in what it consists. It may seem odd to the ear, but the name we give to the study is one which will surprise a person unfamiliar with the subject — astronomy. Are you unaware that the true astronomer, must be a man of great wisdom? I don’t mean an astronomer of the type of Hesiod and his like, a man who has just observed settings and rising, but one who has studied seven out of eight orbits, as each of them completes its circuit in a fashion not easy of comprehension by an capacity not endowed with admirable abilities. I have already touched on this and shall now proceed, as I say, to explain how and on what lines the study is to be pursued. And I may begin the statement thus.

“The moon gets round her circuit most rapidly, bringing with her the month, and the full moon as the first period. Next we must observe the sun, his constant turnings throughout his circuit, and his companions. Not to be perpetually repeating ourselves about the same subjects, the rest of the orbits which awe enumerated above are difficult to comprehend, and to train capacities which can deal with them we shall have to spend a great deal of labor on providing preliminary teaching and training in boyhood and youth. Hence there will be need for several sciences. The first and most important of them is likewise that which treats of pure numbers — not concreted in bodies, but the whole generation of the series of odd and even, and the effects which it contributes to the nature of things. When all this has been mastered, next in order comes what is called by the very ludicrous name mensuration, but is really a manifest assimilation to one another of numbers which are naturally dissimilar, effected by reference to areas….”

Plato presents the irony, of a connection between the study of “pure numbers not concreted in bodies,” and the mastery, in the mind, of the motion of the heavenly bodies — astronomy. As we discovered by our previous investigations into linear, polygonal, and geometric numbers, and Gauss work on the calendar, this connection is in the realm of Higher Arithmetic — Gauss’ re-working of classical science.

In our previous studies, we quickly learned the foolishness of thinking of numbers in connection with objects or bodies. Instead, we began to discover, that knowledge lies in investigating the relations between numbers, not the numbers themselves. We discovered how to begin to distinguish these relations as different {types} of differences (change) among numbers. Numbers, related to one another by the same {type} of difference, are congruent relative to that {type} (modulus). These {types} of differences, can be distinguished from one another, either by magnitudes, as in the case of linear and polygonal numbers, or by incommensurability, as in the case of geometric numbers. As we discovered with the application of Higher Arithmetic to the determination of the Easter Date, when the mind abandons all foolish fixation on objects, and focuses instead on the relations between them, an extremely complex many, can be brought into our conceptual ken.

A similar approach can be taken with respect to the issues Jonathan raised in last week’s pedagogical discussion. Nothing can be discovered about the astrophysical, by, as Plato indicates, simple observations, like the methods of Hesiod. Instead, one must look to the {type} of change, (relations), of which those observations are only a reflection.

Think of two objects, representing two observations of a planet in the sky. What is the relationship between these two objects? What one must be investigate, is the type of difference (change) between those objects. Or, under what curvature (modulus) are the relations between these objects congruent.

For example, if those two objects are related to each other by a straight line, then the type of difference is measured by rectilinear action, no matter how small the interval between them. If, however, they are related to one another by a circular arc, the type of difference will be characterized by constant curvature, not rectilinear action, no matter how small the interval between them. Or, if they are related by an elliptical arc, the type of difference is characterized by changing curvature, no matter how small the interval between them. The mind must distinguish, the type of change, rectilinear, constant curvature, changing curvature, or types of changing curvature. The determination of which type of change, is related to these specific observations, is not a formal question, but a matter of discovery.

By the time he was 16 or 17, Gauss had already discovered a new type of difference, congruence in the complex domain, which he applied to his work throughout his life. Not until 37 years later, in his second treatise on biquadratic residues, did Gauss begin to elaborate the metaphysical principles behind this discovery.

We can gain some insight, into Gauss’ thinking, from the following fragment, taken from one of Gauss’ 1809 notebooks:

Questions to the Metaphysics of Mathematics

1. What is the essential condition, that a can be thought of, to combine concepts with respect to a magnitude?

2. Everything becomes much simpler, if at first we abstract from infinite-divisibility and consider merely discrete magnitudes. For example, as in the biquadratic residues, points as objects, intersections, therefore relations as magnitudes, where the meaning of a + bi – c – di is immediately clear. (This is accompanied by a grid in the complex domain. See “The Metaphysics of Complex Numbers” Spring 1990 21st. Century Magazine)

3. Mathematics is in the most general sense the science of relationships, in which one abstracts from all content the relationships.

Assume a relationship between two things, and call that the simplest relationship, etc.

4. The general idea of things, where each has a two-fold relationship of inequality, are points in a line.

5. If a point can have more than a two-fold relationship, the image of it, is the position of points that are connected by lines, in a surface,. But, if one should investigate here all possibilities, it can only concern the points, which are in a three-fold reciprocal-relationship, and giving a relationship between relationships.

6. It were extremely important, to bring the theory of differences to clarity without magnitudes. As occurs, for example, in the series differences in a plane leveller. The position of the bubble in the glass pipe is determined to be at rest by the geometrical axis of the pipe, and a line through the plane of the feet.

* * *

In this brief fragment, we can see the complete unity in Gauss’ mind, between mathematics, metaphysics, and physics. To help grasp this, the reader should perform the following demonstration with a carpenter’s level, while thinking of the above discussion:

Hold the level on a surface so that the bubble is a rest in the middle. Now rotate the level around a line perpendicular to the surface. The bubble will not move. Now rotate the level along an axis, in the direction of the glass tube. The bubble will still not move. Now rotate one end of the level up and the other end down, on an axis parallel the surface, but perpendicular to the level. The bubble moves. Movement of the bubble back and forth, is inseparably connected with movement of the level in a second direction. These two actions back-forth and up down, are not the same thing in two directions, but One two-fold action.

(If you are self-conscious, while thinking about this demonstration, you should be able to discover where the gremlins of Newtonian mysticism might be lurking in your mind.)

Acutely aware that only metaphor can adequately convey an idea, Gauss wrote to his friend Hansen on December 11, 1825:

“These investigations lead deeply into many others, I would even say, into the Metaphysics of the theory of space, and it is only with great difficulty can I tear myself away from the results that spring from it, as, for example, the true metaphysics of negative and complex numbers. The true sense of the square root of -1 stands before my mind (Seele) fully alive, but it becomes very difficult to put it in words; I am always only able to give a vague image that floats in the air.”

In upcoming weeks, we will re-construct some of Gauss’ metaphors. We leave you today, with the following from the Epinomis:

“Now the proper way is this — so much explanation is unavoidable. To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves, and disclose themselves, they will, if, as I say, a man pursues his studies aright with his mind’s eye fixed on their single end. As such a man reflects, he will receive the revelation of a single bond of natural interconnection between all these problems. If such matters are handled in any other spirit, a ma, as I am saying, will need to invoke his luck.”

## Dance with the Planets

by Bruce Director

Before the excitement from his stunning determination of the orbit of Ceres had begun to diminish, Gauss found, in the tiny variations of the orbit of the newly discovered asteroids, a further means for extending Kepler’s discovery of the non-linear harmonic ordering of the solar system. As those who have worked through Gauss’ method know, the re-discovery of Ceres, in December 1801, accomplished much more than the opportunity to renew observations of Piazzi’s newly found object. Gauss’ accomplishment represented a triumph of Kepler’s successful application of Plato’s and Cusa’s method. Gauss had demonstrated anew what Kepler himself had shown; that the harmonic ordering principles (Kepler’s so-called three laws), existed and could be discovered in every small interval of action in the solar system.

There was another aspect of Kepler’s principles that did not come directly into play in Gauss’ initial determination of the Ceres orbit. In his {Harmonies of World}, Kepler investigated the interaction among the planets themselves, expressed in terms of musical intervals. These intervals were formed by the angular velocity of the planets at their extreme distances from the Sun. It was in these relationships, that Kepler, in fact, had forecast the existence of an exploded planet in the same region of space in which Gauss had determined Ceres to be. Significantly, Kepler noted a small discrepancy, between the calculated values of these angular velocity, and the musical intervals to which they corresponded.

It had since been observed, with the finer measurements of Gauss and his colleagues, that there were small deviations in the observed positions of the planets from purely elliptical orbits, known as perturbations, due to the interaction among the planets themselves.

In the {Theoria Motus}, Gauss says, “The perturbations which the motions of planets suffer from the influence of other planets, are so small and so slow, that they only become sensible after a long interval of time; within a shorter time, or even within one or several entire revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it would not be worthwhile to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been accurately observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion.”

As Gauss indicates, this is another self-reflexive problem: “Since the determination of the elliptic elements with which, in order that the observations may be exactly represented, the perturbations are to be combined, supposes a knowledge of the latter; so, inversely, the theory of the perturbations cannot be accurately settled unless the elements are already very nearly known; …”

The observable effect of these perturbations take two forms. One is the short term perturbations, small deviations from a purely elliptical path, the other is the so-called secular perturbations, that have the effect of changing the size, shape, and position of the orbit itself. In other words, while the planets are moving about the Sun in Keplerian orbits, the orbits themselves are changing positions with respect to each other.

Because these perturbations are so small, they are very hard to measure in the larger planets known before the discovery of Ceres. But, in the much smaller Ceres, and the other newly-discovered asteroid, Pallas, these perturbations were more easily measured. For this reason, Gauss jumped on the opportunity to investigate these tiny changes. By June 1802, Gauss already wrote to his friend Olbers concerning his ideas about calculating the perturbations of Ceres and Pallas. His aim was not simply to better trace the orbital motion of these asteroids, but to investigate a whole new set of cycles that were present in the orbits of every planet. The significance of these perturbation cycles are similar to Pythagoras’ commas or Kepler’s small discrepancy in the musical intervals. Gauss found in these investigations, the means to elaborate a new mathematical metaphor of the arithmetic-geometric mean and hypergeometric functions.

Just as Gauss’ determination of the Ceres orbit itself was a devastating refutation of the fraud of Newton, Euler, and Laplace, the investigation of the perturbation cycles nails their coffins. Kepler had demonstrated that it was the anti-entropic ordering principle of the solar system as a whole, that governed the motion of the individual planets. The oligarchy’s attack focussed on imposing the Newton/Sarpi formalism of the pair-wise inverse square law, that in effect, reduced the harmonic motion of the planets, into a fixed Aristotelian order. A fraud from the beginning that was only maintained through pure political thuggery. Any attempt to calculate the interaction of two planets and the Sun using Newton’s inverse square law, becomes a mathematical impossibility known as the three-body problem. Now, if the interactions of the planets among themselves could no longer be ignored, the whole edifice of Newton’s pair-wise formalism falls apart.

As Kepler showed, the planets are not interacting in a pair-wise manner, but rather comprise a nonlinear harmonic process. However, these harmonic relations are not limited to the motion of the planets around the Sun. To grasp the interaction of the moving planets on each other, while they move about the Sun in Keplerian orbits, requires the mind to be able to think in terms of cycles moving cycles, or modular functions.

One may feel a tiresome tug in response to this thought. When do these cycles ever end? Is there not one cycle of cycles, by which all these motions can be brought into a One? Each new discovery, seems to leave something incomplete. None of the cycles ever close.

To get in the right frame of mind to approach these questions, it is appropriate to look to a passage from Friedrich Schiller’s Aesthetical Lectures:

“I know of no more suitable image for the ideal of beautiful behavior, than a well-performed English dance, composed from many complicated figures. A spectator from the gallery sees innumerable movements, which cross one another most vividly and alter their direction briskly and playfully, and yet never knock into one another. Everything is so ordered, that the one has already made room, when the other arrives; everyone fits so skillfully and yet again so artlessly into one another, that each seems to follow only his own head and yet never steps in the way of the other. It is the most suitable emblem of the asserted self-freedom and the spared freedom of the other.”

To dance with the planets, one must think of a dance in which the steps of the dancers alter the curvature of the dance floor, which in turn changes the motions of the dancers.

Gauss developed a beautiful approach to this dance, that we’ll examine next week.

To What Do the Planets Dance?

by Bruce Director

The motion of the newly-discovered asteroids, Ceres and Pallas, presented to Gauss a renewed opportunity to advance the Platonic method of astrophysical investigation that Kepler had presented to the world nearly 200 years earlier. In the intervening 200 years, the oligarchy, through their Leporellos, Euler and Newton, had perpetrated a hoax, to undo Kepler’s accomplishment, not by locking up what Kepler wrote, but by locking up the minds of those who would read it, and not wish to be on the wrong side of popular opinion. Gauss, with his determination of the orbit of Ceres, wielded the power of the universe against those oligarchical mind games, and, like a good commanding general, pursued the enemy, with the investigation into the tiny perturbations of the Ceres and Pallas orbits.

This oligarchical mind control exists to the present day. Last month, Roger Ham forwarded a copy of the Summer Fidelio to one Brian Marsden who specializes in the determination of orbits at the Smithsonian Astrophysical Observatory of Harvard University. Within 3 days time, the professor replied, “I fully agree that Gauss’ method of orbit determination has great elegance and power, and we certainly make use of it in our own work, including what we did on 1997 XF11. Nevertheless, that is not the whole story, and it was quite improper of the Fidelio article to criticize Newton and Euler in the way it did….”

Another professor, MIT’s Laurence Taff, displays a similar ignorance in his 1985, text “Celestial Mechanics–A Computational Guide for the Practitioner.” In the section on the determination of planetary orbits, Taff quotes extensively from Gauss’ Theoria Motus. Then, after a brief description, albeit in formal terms, of Gauss measurement of non-linear curvature in the small, Taff states, “Although this picture of the Gaussian process is interesting, it is also irrelevant.” Under the heading, “The Historical Myth–Ceres,” Taff says, “We do not know how Gauss actually computed the orbit of Ceres or any of the other big four minor planets. We do know that however he did it.”

Both men trumpet the superiority of the statistical methods, rejected by Gauss, because those methods are more practical, especially with the use of high speed computers. Their obsession, however, is not driven by the quality of their results, but by the emptiness of their minds. To them, there are those methods of elegance and power, and then there are the not “improper” practical ones.

In this light, let’s continue to look at Gauss’ investigation of the perturbations of the orbit of Ceres and Pallas, from Gauss’ standpoint.

As Kepler, following Plato and Cusa, demonstrated, the solar system is a harmonically-organized process. For the mind to grasp that harmony, one must be able to “hear” the polyphony of interactions among all the heavenly bodies. Kepler presented that polyphony to us, in the form of his three principles of planetary motion, and the musical relationships of the planet’s extreme angular velocities.

If, however, when you think of Kepler’s solar system, you think of each planet, separately moving about the Sun, you are already on the slippery slope into the Sarpi/Newton abyss. It is but a short step, in this mode of thinking, to translating Kepler’s principles, by algebraic manipulation, into a seemingly more practical system of formal equations such as Newton’s inverse square law. Will you fall into the trap, that Kepler’s principles are of “elegance and power,” but Newton’s laws say the same thing in a way more suitable for calculation as, Cal Tech’s Richard Feynman, who, in his famous “lost lecture,” committed the sophistry of pretending to prove “geometrically” that the planets obey Kepler’s principles as a consequence of the inverse square law?

Nothing strikes terror into the mind of the formalist, more than the elegance and power of the Socratic method, just as Gauss and Kepler to this day strike terror in the minds of Marsden, Taff, and Feynman. That terror grows stronger when confronted with the physical evidence that the planets don’t obey the formalist’s rules. If Kepler’s and Gauss’ work on the larger motions of the planetary orbits were not enough, the whole Newtonian edifice, and the mental states of its adherents is completely thrown into disarray, when the tiny perturbations of the planetary orbits are considered. As Taff admits in the above mentioned text, “In my view, the problem is that when you really need perturbation theory (in celestial mechanics) it does not work very well (as applied in the past). The existing applications of it do not explain the Kirkwood gaps in the asteroid belt. They do not explain the fascinating discoveries in the Saturnian rings/moon system. They do not predict physically correct rates of change of the orbital elements of a satellite revolving about an oblate primary. Some of these failures may be attributable to faulty mathematics, some to the difference between a trajectory and a field theory. The reader is advised that the above is a minority opinion (and taken slightly out of context).”

Taff’s uneasiness, once again, comes from his obsessive devotion that has lead him, like the Harvard professor, to turn away from methods of power and elegance, for the more practical ones.

The problem is straight forward. All the heavenly bodies that move about the Sun are moving in Keplerian orbits, and, these orbits, as Kepler demonstrated in the Harmonies, are interacting with each other. The effect of these inter-orbital interactions, is measurable in terms of small variations in the orbital elements that characterize the size, shape, inclination, and position, of the orbits. In other words, the planetary orbits are not fixed tracks in the sky, but regions of motion, in which the planets move, similar to the regions of ambiguity that characterize the tones of the well-tempered system of polyphony. The curves traced out by these motions, therefore, are not the simple curves you draw on a piece of paper, but much more complicated. (The reader can make a representation of this, by drawing a circle with a compass and slowly making the radius slightly larger and smaller as you rotate the compass around the center.) This representation is merely the footprint of the planet’s motion. Now think of how this type of motion would be reflected in changes in the orbital elements.

The measurable variations of these orbits occur in very long cycles, but, these cycles are very significant. For example, the precession of the equinoxes is one effect of these perturbations. A longer 100,000 year cycle in the eccentricity of the Earth’s orbit, resulting from these orbital interactions, accounts for the cycle of ice ages.

The scientific question posed, is, what is the harmonic relationships that are reflected in these inter-orbital interactions? In the larger planets, as Gauss says, these cycles are so small and slow as to be nearly insensible, over short intervals of time. But, in the orbits of the smaller planets like Ceres and Pallas, these cycles are measurable during shorter time intervals, and in them, the underlying harmonic ordering principle of all the inter-orbital interactions are potentially discoverable.

How do we measure the interactions between orbits?

In an 1819 paper, “The Determination of the Attraction of an Elliptical Ring,” Gauss discusses the following principle of physical astronomy:

Changes in the elements of a planet’s orbit, that are due to the interaction with another planet, are independent of the position of the disturbing planet. That is, the long range changes in the size, shape, inclination and position of a planet’s orbit, that arise from the interaction with another planet, arise from the interaction of the two orbits. Furthermore, Gauss says, that if the total time it takes for both planets to make one complete revolution around the Sun, are incommensurable, then the effect of the disturbing planet on the elements of the disturbed planet, can be calculated using the following metaphor:

Think of the mass of the disturbing planet as spread out in a ring of uniform, but infinitely small thickness, such that the mass is distributed in this ring, proportional to the speed of the planet. That is, since the planet spends more time in the parts of the orbit farther from the Sun, and less time in the parts of the orbit closer to the Sun, and, according to Kepler’s area principle, equal times sweep out equal areas, any piece of the elliptical ring, in which the planet spends equal times, will have equal amounts of mass. Or, in other words, the mass of the planet should be thought of spread out non-uniformly, in proportion to, the non-uniform motion of the planet.

The problem Gauss then solved, is how to determine the effect of this elliptical ring, on any other position in space. This lead Gauss to a surprising application of the arithmetic- geometric mean and a deeper insight into modular functions.

For now, however, it were useful to turn again to Friedrich Schiller, whose poem, “The Dance” presents to us the appropriate conceptions for the task.

See how with hovering steps the couple in wavelike motion
Rotates, the foot as with wings hardly is touching the floor.
See I shadows in flight, set free from the weight of the body?
Elves in the moonlight there weaving their vapor-like dance?
As by zephyr ’twere rocked, the nimble smoke in the air flows,
As so gently the skiff pitches on silvery tide,
Hops the intelligent foot to melodic wave of the measure,
Sweet sighing tone of the strings lifts the ethereal limbs.
Now, as would they with might traverse through the chain of the dances,
Swings there a valorous pair right through the thickest of ranks.
Quickly before them rises the path, which vanishes after,
As if a magical hand opens and closes the way.
See! Now vanished from view, in turbulent whirl of confusion
Plunge the elegant form of this permutable world.
No, it hovers rejoicing above, the knot disentangles,
Only with e’er-changing charm rule does establish itself.
Ever destroyed, creation rotating begets itself ever,
And an unspoken law guides the transformative play.
Say, how’s it done, that restless renews the swaying formations
And that calmness endures even in moveable form?
Is each a ruler, free, to his inner heart only responding
And in hastening course finds his own singular path?
Wish you to know it? It is the mighty Godhead euphonic
Who into sociable dance settles the frolicking leap,
Who, like Nemesis fair, on the golden rein of the rhythm
Guides the raging desire and the uncivilized tames.
And do the cosmos’ harmonies rustle you to no purpose,
Are you not touched by the stream of this exalted refrain,
Not by the spirited pulse, that beats to you from all existence,
Not by the whirl of the dance, which through eternal expanse
Swings illustrious suns in boldly spiraling pathways?
That which you honor in play–measure–in business you flee.

Translation by Marianna Wertz

A Pedagogical Example, Designed to Perturb the Mind, With a Problem of Non-Linearity

by Bruce Director

When Kepler discovered the ordering principle of planetary orbits — that the non-uniform motion of a planet can be measured by the area swept out by a line connecting the planet and the sun, such that those areas are equal for equal intervals of time elapsed — he immediately confronted the problem of mathematical physics identified by Nicholas of Cusa in “On Learned Ignorance.”

If you look on p 28 of the Fidelio article on the determination of the orbit of Ceres, you can see in Fig. 5.5 that the area swept out by the planet’s motion from P1 to P2 (the curvilinear area P1-A-P2 shaded in white) is measured by the length of the line P1-N, which is the sine of the angle P1-B-P2. As the quote from Kepler on that page indicates, the ratio between the arc and the sine are infinite.

Kepler called for the discovery of the calculus to solve this problem. Leibniz made the initial discovery, and Gauss, extending the discoveries of Leibniz and their Greek predecessors, took it to a new dimension with his investigation of the hypergeometric function and the arithmetic-geometric mean.

Look again at the problem in such a way as to bring out the underlying characteristic of action. Referring again to Fig. 5.5 on p 28, think of the type of change of the right triangle P2-N-B, as the planet moves counter-clockwise around the circle, from P1. The hypotenuse of the triangle, P2-B remains constant, but the lengths of the legs change, with each change in the angle, P1-B-P2. If we consider the hypotenuse to be 1, then for any given angle, we can calculate the lengths B-N, (cosine) and the P2-N (sine), by the methods Euclid used to divide the circle.

For example, dividing the circle into 4 parts, would create angles for P1-B-P2 of 0, 90, 180 and 360 degrees. The magnitudes of the lengths of B-N (cosine) would be 1,0,1,0 respectively and the magnitudes of the lengths P2-N (sines) would be 0,1,0,1 respectively. (I have left out considerations of direction for the moment.)

Now, dividing the circle again into four more parts, adds the angles, 45, 135, 225, and 315 to the previous four angles. Now, to calculate the corresponding lengths for B-N (cosine) and P2-N (sine) we can use the Pythagorean theorem. Since by definition the angle P1-B-P2 is 45 and the angle P1-N-B is 90 degrees, the remaining angle in triangle P2-B-N is also 45 degrees making that triangle isosceles. Since the hypotenuse P2-B is 1, the lengths of the sides, will each be equal to 1/2 times the square root of 2.

To continue this process, we are at first limited to those divisions that can be accomplished by circular action itself, the so-called straight edge and compass constructions. From Euclid’s time until Gauss, those divisions were limited to 3, 4, 5 and certain combinations and multiples of those divisions. Greek geometers, such as Hippias, discovered that using other curves, such as the quadratrix, the circle could be divided in other ways. In 1794, Gauss discovered a heretofore undiscovered ordering principle of circular action, by which the circle could be divided into 17, 257, and other parts.

Furthermore, the sine and cosine of 45 degrees, 1/2 the square root of 2, while constructable with straight edge and compass, is nevertheless and irrational magnitude. For the sines and cosines of other angles, non-constructable irrational magnitudes arise.

Something else can now be brought into view; the underlying non-linearity of the transcendental relationship between the circular arc and the straight line. When we divided the circle into 8 parts, the angles were divided equally. That is, 45 degrees is exactly half of 90 degrees. But, when we calculated the lengths of the sine and cosine, they did not change by 1/2. In fact, the sine (length P2-N) more than doubled, from 0 to 1/2 times the square root of 2, and the cosine (length B-N) decreased by less than half, from 1 to 1/2 times the square root of 2. If we were to calculate the sine and cosine for other angles, we would see that the sine and the cosine change non-uniformly as the angle changes uniformly.

This should surprise you. The circle is a curve of constant curvature, yet the transcendental relationship between the arc and the straight lines is non-constant! This is nub of the problem that Kepler confronted. To solve it, one would have to get at the underlying non-linear relationship between arcs and straight lines.

This is a classic example of inversion. If we change the angle by a given interval, the sine and cosine change by a different interval. As in the above example, if the angle changes from 0 to 45 degrees, the sine changes from 0 to 1/2 times the square root of 2. But, if the angle changes from 45 to 90 degres, a uniform change in angle, the sine changes by a smaller interval.

So in this direction we ask, “Given a change in angle, what is the resulting sine and cosine?” But the inverse question, “Given a change in sine, how much does the corresponding angle change?” The determination of the first question is difficult, the second, (because the sines are incommensurable magnitudes) seems impossible, unless, we can discover some unseen relationship that governs this non-linear process.

Now, in the case of the elliptical orbits, the problem seems even more complicated, because the ellipse is a curve of constantly changing curvature. In an ellipse, not only do the legs of the triangle P2-N and N-B change non-linearly, but also the hypotenuse P2-B. (For this we refer to figure 6.4 on page 33 of the Fidelio.) Using the principles of conical proportions discovered by Appolonius of Perga (A contemporary and associate of Eratosthenes, Aristarchus and Archimedes), Kepler avoids the problem of calculating the elliptical sector by calculating the proportional circular one. (Q-A-N in figure 6.4 This is done by the same method as illustrated on page 28.)

In the determination of the attraction of the elliptical ring, Gauss, instead of trying to solve the difficult problem, took on the seemingly impossible one, finding, at least to a certain degree, the unseen relationship that governs the non- linearity between the angle and the sine and cosine.

For this, Gauss began with a sort of inversion of Kepler’s eccentric anomaly. If we think of the ellipse in figure 6.4 as the elliptical ring of the disturbing planet, the disturbed planet would lie along line B-P2 (not drawn in the figure). In other words, instead of thinking of P2 as the position of the planet in motion, we should think of a planet moving inside the ring. P2’s position is the “projection” of the motion of the disturbed planet on the elliptical ring. The position of P2, therefore changes with the motion of the moving planet. Then, Gauss “projected” P2’s position to Q. This enables Gauss to measure the non-uniformly changing length B-P2, as a function of angle E (the eccentric anomaly.)

(What is a work here, is the method of interconnected colligating cycles, in which we are interested in the way changes in one cycle are conncect to changes in other cycles. It is the type of changes we are investigating.)

Now applying the principle of inversion, Gauss seeks to determine the characteristic of change between the straight line B-P2 and the angle E. Instead of trying to calculate the actual change in length of B-P2 with respect to any particular change of angle E, Gauss seeks to determine the characteristic form of the change, as E sweeps around one entire revolution.

Beginning with Appolonius’ properties of the ellipse, Gauss shows that the relationship of B-P2 to angle E, is a function of the semi-axes of the ellipse. Specifically, if the semi-major axes of the ellipse is A and the semi-minor axis is B, then P2’s position is determined by A times B-N (cosine of E) and B times Q-N (sine of E). Another way to think of this, is that the ellipse in figure 6.4 is a contraction of the circle, and that P2’s position is formed by contracting length Q-N by a factor of B/A. (For a discussion of the basic relationships of the ellipse see the Appendix to the Fidelio article.)

Then Gauss determines that the characteristic change of B-P2 with respect to E, is invariant under the following transformation:

If A is the semi-major axis and B is the semi-minor axis, of the ellipse, calculate the arithmetic mean (A+B/2) of these magnitudes, and call that A’. Then calculate the geometric mean (the square root of AxB) call that B’. Then construct a new ellipse with semi-major axis A’ and semi-minor axis B’. Then repeat this process. This construction, leads to a discontinuous sequence of ellipses, the semi-axes of each ellipse, are the arithmetic and geometric means of the previous one. This series converges rapidly on a circle, whose radius is the arithmetic- geometric mean.

(A graphic representation of this process can be seen on page 51 and 52 of “So, You Wish to Learn All About Economics?”)

What Gauss demonstrated, was, that the characteristic change of the inverse of the length of the line B-P2 with respect to angle E was the same for each ellipse in this series. And, if calculated for an entire circuit around the ellipse, is equal to 1/arithmetic-geometric mean of the semi-axes of that ellipse.

Once again, Gauss has presented us with a paradoxical solution, for the arithmetic-geometric mean, is itself a kind of transcendental and so we are measuring one transcendental by another. Gauss went on to discover that these processes are related as part of a whole group of transcendental processes subsumed under his conception of hyper-geometric functions.

In the announcement of his study on the hyper-geometric series, Gauss said, “The logarithmic and circular functions, as the simplest manner of transcendental functions, are those, with which analysis has been mostly occupied. They earn this nobility because of their continuous hold on almost all mathematical investigations, theoretical and practical, and because of their almost inexhaustible richness of interesting truths that their theory presents…” Gauss goes on to discuss the other higher forms of transcendental functions saying, “The transcendental functions have their source at all times, often reclining or hiding, in the infinite …” These hidden sources, Gauss explores in the domain of hyper-geometry.

## What Counts or How Your Days Are Numbered

By Bruce Director

Most people, because of an apparent childhood recollection, associate whole numbers with the counting of objects, and, as such, the numbers 1, 2, 3, … to whatever large number imaginable seems to be the natural order of things, hence the name “natural numbers.” But as St. Paul says in the oft-cited 1 Corinthians 13, “When I was a child, I spake as a child, I understood as a child, I thought as a child; but when I became a man, I put away childish things.” And Plato in the Republic teaches that for warriors to become mature leaders, they must learn to “see the nature of numbers with the mind only”. In the following weeks’ pedagogical discussions, we will begin to learn to think of the “nature of numbers,” instead of “natural numbers”.

The association of whole numbers with the counting of objects seems plausible in the small, but, as we begin to think of larger numbers, the association with the counting of objects seems less and less valid. It is clear we think not in terms of single numbers, but in terms of groups of numbers, say groups of 10. When we reflect back, we realize that even with small numbers this is the case, and what had seemed plausible, was wrong.

In “On Conjectures” Cusa discusses the “Natural Progression” of numbers this way:

“To contemplate the nature of number is more acutely useful to you, the more deeply you attempt to investigate the rest in its similitude.At first, however, concern yourself with its progression, and you shall confirm that it is accomplished by the quaternary. Indeed, one, two, three and four added together produce ten, which unfolds the natural power of simple unity.mNow from this same ten, which is the second unity, the quadratic unfolding of the root is achieved through a similar quaternary progression: 10, 20, 30 and 40 added together are one hundred, which is the square of the denary root. Likewise, the hundred exerts the thousand as unity through the same movement: 100, 200, 300, and 400 added together are one thousand. Yet do not proceed further on this path, as if something still remained.

“However, not only after the ten–as with eleven, where after the ten a regression to unity occurs–but also in a similar manner after the thousand, the repetition is not denied; in the natural influx, there are therefore no more than ten numbers, which are contained in a quaternary progression. And beyond the one thousand, the sum of the cube of the denary root, there is no variation in the repetition, since this arises through the triply repeated quaternary progression in the denary order. Consider also, that the quaternary, the unfolding of unity, contains the power of the total number….”

We move closer to knowing the “nature” of numbers, when we think of larger numbers, such as the number of years in the procession of the equinox (The change of the constellation of the zodiac which rises with the sun at the equinox.) This is an approximately 26,000-year cycle, reflecting the juxtaposition of the cycle of one rotation of the earth around the sun, and one rotation of the direction of the earth’s axis with respect to the stars. Both cycles are seen in the mind only, for we only see the changes in position of the sun in the sky, and the changes in position of the stars. We never see the actual motion of the earth around the sun, or the rotation of the earth’s axis itself. When we juxtapose in our mind, these two mental concepts, the number 26,000 arises. No one has ever, nor could ever, arrive at this number by counting. The number arises solely from the construction of a metaphor, composed by juxtaposing two distinct ideas, associated with physical processes.

All numbers we can think of, are associated with metaphors in a similar way, as the metaphor of Gauss’ Easter algorithm beautifully illustrates. No number is ever thought of by itself, but only in relation to some metaphor, even though the underlying metaphor is not always apparent. To discover the deeper implications of this fact, we should try some experiments, using Gauss’ “Disquisitiones Arithmeticae,” as our guide.

First, we must abandon all together, any mental dependence on the “natural” order of whole numbers. (This is easier said than done, as this childish idea–which doesn’t come from children–is a very deeply embedded axiomatic assumption.) Instead, think of numbers with respect to Gauss’ concept of congruence.

If the difference between two numbers is divisible by a third number, they are said to be congruent with respect to the third number. The two numbers are called residues of each other, the third number is called the modulus.

Sound simple? Try some examples. I’ll give you a couple to get you started. 13 and 138 are congruent relative to modulus 5, but non-congruent relative to modulus 11. -9 and +19 are congruent relative to modulus 7.

Under this concept of congruence, the relationship of three numbers, defines the relationship of all numbers to each other. No number is self-evident, but are related to one another by the relationship of the {interval} between them to the modulus. Each interval, defines under what moduli, these numbers are congruent or non-congruent. This concept of number, as the juxtaposition of two ideas, an interval and a modulus, more truthfully reflects the way numbers actually arise in the mind.

By a simple experiment, one can see how deeply embedded the so-called natural order of numbers is. Think of any number as a modulus, called m. Now take a series of numbers, begin with any arbitrary number, a, and write down a sequence of numbers a, a+1, a+2,…a+m-1. The number of numbers in this sequence, will equal the number of 1’s in the modulus. Now take another number not in the sequence, called A. This number will be congruent to only one number in the sequence, relative to the modulus m. Now take A+1. What number is that congruent to? Then A+2, until you get to A+m-1. See what happens when you continue to add 1 to A after this point.

Try this with different numbers for a modulus. Does your naive imagination insist on thinking of the natural sequence of numbers a, a+1, a+2, … as primary, and the concept of congruence as intrusive? Do you find yourself becoming angry at the way the concept of congruence disrupts the simple “natural” ordering of whole numbers? If, as we illustrated above, that in the real world, all numbers arise from metaphor, why is your mind so insistent on protecting, the seemingly childish idea of the natural ordering of numbers.

Now have some more fun. Instead of adding 1 to a, add 2, or 3, or 4. See what happens.

Recently a good friend, who resides temporarily in Baskerville, Va. suggested the following real life example of counting by congruence with respect to a modulus, known as the “Baskerville Modulus Problem”. In a prison with three buildings, each with two dorms A and B, each dorm rotates going to lunch first. If dorm A in Building 1 goes to lunch first on Monday, how long will it be before that same dorm eats lunch first on a Monday? Once you figure that out, chicken is served every third Thursday. If the dorm doesn’t get called first, the chicken gets cold and rubbery. How often does dorm A of Building 1 get hot chicken?

Don’t Count Your Chickens, Unless They’re Hot

We left you last week in the mind of a prisoner contemplating the timing and quality, (or lack thereof) of his future meals, so as to confront the contradiction between the naive imagination’s belief in the “natural progression” of numbers, and, the “nature” of numbers, as seen with the mind only. If you took into your mind the prisoner’s plight, the real life implications of this contradiction should have become clear to you.

In the first part of the “Baskerville Modulus Problem”, our prisoner is caught in a six-day cycle, in which he goes to lunch first once every six days. Count this rotation according to the “natural” order of numbers, 0, 1, 2, 3, 4, 5, with 0 counted for eating lunch first, 1 counted for eating lunch second, etc. But, counting according to the “natural” order of numbers, and, the reality of the prisoner’s lunch schedule begins to diverge on day 6, for on the sixth day, the prisoner eats lunch first again, counted as 0 in the order of eating lunches. On the seventh day, the prisoner eats lunch second, or 1 in the order of eating lunches, and so forth.

Now think of this in terms of Gauss’ conception of congruence. On day 1, the prisoner eats lunch first–0 in the order of lunches. On day 6, the prisoner eats lunch first again–0 in the order of lunches. So, there is something the same between 1 and 6. Obviously, they are not equal, but they are {congruent} relative to modulus 6. Because the interval between 0 and 6 is divisible by 6. (Gauss introduced a new symbol to distinguish congruence from equality. That symbol is an equal sign with an extra line. Due to technical considerations we can not reproduce that symbol here.)

Continuing on to the 12th day, the prisoner eats lunch first again. So, 0 is congruent to 6 and 12 relative to modulus 6. Again, the intervals between 0 and 6, 6 and 12, and 0 and 12 are all divisible by the modulus, 6. Similarly on day 7, the prisoner eats lunch second (counted 1). 7 and 1 are not equal, but are congruent relative to modulus 6. This relationship, of congruence with respect to modulus 6, reflects the specific ordering principle, of the domain of the prisoner’s lunch schedule, which diverges from the simple linear counting of the days according to the “natural progression” of numbers.

This concept of congruence, is analogous to Kepler’s concept of congruence in the first book of the Harmonies of the World. There, Kepler notes that the word congruence means to Latin speakers what the word harmony means to Greek speakers. The Greek word for harmony and arithmetic, come from the same Greek word-stem which means, in English, “to fit together.” Kepler denotes as congruent, polygons which can be fitted together. For example, in a plane, triangles, squares, and hexagons, can be fitted together perfectly, while pentagons cannot. On the other hand, in a solid, triangles, squares, and pentagons, can be fitted together, while hexagons cannot. What polygons are congruent and what are not, is dependent on the domain in which the congruence is formed.

This relationship of congruence with respect to domain, can be seen in our prisoner’s situation. There are 7 days of the week, which can be counted according to the “natural progression” by Sunday=0, Monday=1, etc. Here again, the “natural progression” and reality diverge when we get to the second Sunday, which is the 7th day (since we started counting with 0 not 1). Again, this second Sunday, is similar to the first, but it is obviously not equal, it is {congruent}–0 is congruent to 7 modulus 7.

The days of the week are ordered according to congruences relative to modulus 7 and the prisoner’s lunch schedule is ordered according to congruences relative to modulus 6. The numbers 0-6, relative to modulus 7 and 0-5, relative to modulus 6 were called by Gauss the “least positive residues” of the modulus, because they are the smallest positive numbers, with which every other number, no matter how large, will be congruent.

Now combine these two ordering principles. The prisoner’s eating schedule (modulus 6) with the days of the week (modulus 7). If, the prisoner eats first on Monday (counted as day 1) how long will it be before he eats first again on a Monday? If you worked through this problem after last week’s pedagogical discussion, you would have found, that our prisoner ate lunch first on Monday after 42 days (6×7) or every 6 weeks, and, in the intervening period, the prisoner ate lunch first once on every day of the week.

But now look at the second part of the problem: Chicken is served every third Thursday, and if the prisoner doesn’t get there first, the chicken is cold. How often does he get hot chicken? Count the weeks, 0, 1, 2, 3 etc. On week 0, chicken is served on Thursday, on week 1 and 2 no chicken is served, but on week 3 chicken is served again. 3 and 0 are congruent relative to modulus 3. So, chicken is served on those Thursdays which fall on weeks that are congruent to 0 relative to modulus 3.

What does this mean for our prisoner? Well, since he eats lunch first on the same day of the week, once every 6 weeks, he’ll eat lunch first on every sixth Thursday. Or, on those Thursdays which occur on weeks whose numbers are congruent relative to modulus 6. So he’ll get hot chicken every 6 weeks. Right? Not necessarily. If he happens to be lucky enough, that he eats lunch first on a Thursday that serves chicken, he’ll get hot chicken every 6 weeks, but, if on the Thursday he eats lunch first, no chicken is served, he’ll never get hot chicken. In fact, only two groups of prisoners will ever get to eat hot chicken.

What has happened to our numbers? First, the natural progression diverged from reality, but we discovered a more real ordering principle. But now it seems to diverge again. There must be another ordering principle involved, of which we were unaware.

Look back at the difference between the two parts of the problem. In the first part, we combined two cycles, one relative to modulus 6 and one relative to modulus 7. In the second part, the two cycles were relative to the moduli 6 and 3. What’s the difference?

One way to discover from whence this paradox arises, is to shift gears, in a direction discussed by Larry Hecht in a pervious pedagogical discussion concerning the discovery of Gauss’ contemporary Louis Poinsot. (See Pedagogical Discussion 97136lmh01). Think of 7 points (or vertices of a polygon) placed roughly on a circle. Number those points, 0-6. Now connect the points in order 0, 1, 2, 3, 4, 5, 6, 0. Now connect the points, skipping every other one: 0, 2, 4, 6, 1, 3, 5, 0; then skipping every third one; 0, 3, 6, 2, 5, 1, 4, 0; and so forth, for all the numbers 0-6. (This is best done using a different 7 points for each drawing, rather than super-imposing each one on top of each other.) What are the differences and similarities between each figure created by this method? Now do the same with 6 points. Compare these orderings, with the sequences of least positive residues of the series: a, a+1, a+2…; a, a+2, a+4, a+6…; a, a+3, a+6, a+9,…; a, a+4, a+8, a+12…; etc. with respect to modulus 6 and 7.

After performing the above experiment, we can now see more clearly the paradox of the “Baskerville Modulus Problem.” The ordering principle, of which we were previously unaware, is the concept of prime numbers. (See Pedagogical Discussions 97056bmd02 & 97066bmd001.) The number of vertices, is one modulus, and the number skipped, is another modulus. When the two moduli are combined, a polygon is created. The shape of the polygon, and the order in which the points are connected, is dependent on which two moduli are combined. In the case where the two moduli are prime relative to one another, (i.e., they have no common factors), all the points are connected, and a complete polygon is formed. In the cases where the moduli are not prime relative to one another, such as 3 and 6, the polygon will not be complete, leaving some vertices unconnected.

So unless, the modulus of the lunch schedules, and the modulus of the schedule when chicken is served, are relatively prime, there is no guarantee that our prisoner will get hot chicken.

What the above demonstration proves, without a doubt, is, that, only by freeing our minds from the prison of linear thinking, and effectively organizing so as to bring about the complete exoneration of LaRouche, can we insure that our prisoner will get hot chicken, whenever he wants it!

Thank God for the Odd One-

by Bruce Director

We last left our prisoner confronting the divergence of, on the one hand, the endless succession of days, one after the other, and, on the other hand, the actual ordering of those days according to the physical events that occur in them. This clash, between two concepts of number, sparks our prisoner to embark on a journey to discover the nature of number, beyond the realm of sense-certainty. Reflecting back on his childhood education, he realizes that his thinking about number is confined to a rigid set of rules and operations, mere manipulations of numbers as external objects, memorized, not discovered, to be re-called on command.

Suddenly the liberating words of Nicholas of Cusa from {On Conjectures} come to his mind: “The essence of number is therefore the prime exemplar of the mind…. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the Divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

He begins to recall some happier memories of his childhood quest for knowledge, reminiscing how he once playfully discovered hidden relationships among numbers, while secretly exploring their nature with his mind only. Little things, oddities he kept to himself. Once, he had ventured to tell his teacher about one such discovery, only to be discouraged by the response, “Don’t be an oddball. That has no practical application. You won’t need that in later life.”

Now such canons and dogmas memorized in youth are of no use, if they ever were. He finds himself free to inquire anew, beginning first with those elementary principles, which, never simple (except to the simple-minded), unfold a rich bounty of profound ideas, if the underlying, seemingly subtle, paradoxes are sought out.

He takes out a paper and pencil, and unfolds a series a numbers with the following construction:

Begin with a unit *

and add a unit **

and add another unit ***

and another unit ****

and so forth…. *****

******

*******

********

*********

**********

***********

************

It seems apparent enough, from the method of construction, that each number is unique, differing from all others by its relationship to the process of adding one, just as each day follows another. But, seeking to shed the shackles from his mind, our prisoner tries to discover what is behind the numbers, by looking into the numbers on a different level, besides the succession of adding one. He tries the following experiment:

With each number, he alternately marks one unit from each end, beginning with the first and last unit, then proceeding to the second and second to the last unit, until he can go no further.

(The reader is required to make his or her own drawings by hand, rather than rely on computer generated images. Hand drawings, even crude ones, contain within them the cognitive process, whereas the computer images suppresses same.)

What emerges, from this process, is that numbers are distinguished from one another by more than just adding one. Some numbers, (every other one) has a unit left unmarked in the middle, while in the others, no unit remains in the middle.

Again the words of Nicholas of Cusa come to mind: “It is established that every number is constituted out of unity and otherness, the unity advancing to otherness and otherness regressing to unity, so that it is limited in this reciprocal progression and subsists in actuality as it is. It can also not be that the unity of one number is completely equal to the unity of another, since a precise equality is impossible in everything finite. Unity and otherness are therefore varied in every number. The odd number appears to have more of unity than the even number, because the former cannot be divided into equal parts and the latter can be. Therefore, since every number is one out of unity and otherness, so there will be numbers in which the unity prevails over the otherness, and others in which the otherness appears to absorb the unity.”

A smile comes across the prisoner’s face as he now sees the once familiar concept of even and odd numbers (thrust at him as an almost trivial distinction in his youth), in a new light. His joy is mixed with a tinge of anger, as he realizes this new light is not new at all, but, in fact, an ancient discovery, he should have relived as a youth. Unlike what he was taught in school, the concept of even and odd, is not a mere description about a particular number, but a concept associated with the {nature} of number itself. The doctrine he was taught in school seemed to work, but because of it, his mind didn’t.

His anger abates as he turns back to his inquiry. He leaves it to others to uncover how these ancient discoveries were written out of the curriculum.

The infinity of all numbers, has now been divided by two, according to the nature of the individual numbers, when each of them is divided by two. The discovered principle of even and odd, divides the infinity of numbers into two {types} — those numbers in which “otherness prevails over unity,” and those numbers in which, “unity prevails over otherness.”

From the original construction of all numbers by adding one, no number is equal to any other number. But now, he discovers some numbers are alike but not equal to others. To bring this discovery into a One, the prisoner is taken to Gauss’ concept of congruence. All numbers of the same {type} are congruent to each other, and those of a different {type} are non-congruent. There are two {types}. So under Gauss’ concept, all even numbers are congruent to each other relative to modulus two. Likewise, all odd numbers are congruent to each other relative to modulus two. And, all even numbers are non-congruent to all odd numbers relative to modulus 2.

Seeing this, the prisoner desires to continue the exploration, dividing the numbers again. This time, he starts with the even numbers only, taking the parts created from the first division, and marking off the units from each end until he can go no further. (The reader is required to complete this step for himself.)

Now the even numbers have been divided into two {types}; those whose parts when divided in this way, leave no unity — called even-even, and those whose parts, when divided this way, still have a unity in the middle of the part — called even-odd. The odd numbers, in turn, are divided into two {types}. Those which have even numbers on each side of the unity left in the middle — called odd-even and those which have odd numbers on each side of unity that was left in the middle — called odd-odd. The infinite has now been divided four times!

Again, numbers of the same type are not equal, so we go to Gauss’ concept of congruence, to bring this new discovery into a One. Each number of any of these four types — even, odd, even-odd, even-even, odd-even, odd-odd, is congruent to all other numbers of that type, relative to modulus 4.

Yet there is nothing self-evident, from the construction of numbers by the addition of one, from which the now-discovered distinction between even, odd, even-even, even-odd, odd-even, and odd-odd, logically follows. To be able to envisage, from this small distinction among numbers, a different domain, other than the linear domain of adding one, the prisoner must free himself from the constraints of his formal thinking. That domain is characterized, not by linearity, but by curvature, of which the principles of even and odd are but a reflection. The nature of that curvature will be further discovered, by new investigations to which the prisoner looks forward.

As the prisoner contemplates his next experiments, he’s interrupted. Oddly enough, it’s time for lunch.

The Prisoner and the Polygon

Back from lunch, our prisoner eagerly digs deeper into his investigations of the nature of number, fueled by enthusiasm from his recently demonstrated capacity to discover truth by his own powers of reason. He’s determined to avoid the various textbooks lying around (not wanting to fraternize with the enemy), relying instead on a well-worn copy of Euclid’s Elements, whose text contains the footprints of some classical Greek discoveries. The more profound nature of these discoveries are not explicitly stated in Euclid’s Elements, but the profound nature of these ancient thoughts are, neverthelss, reconstructible in the mind. Combining centuries of discoveries in his mind simultaneously, he turns to Book 9, Propositions 21-34 to reconstruct for himself, the indicated discoveries concerning even and odd numbers, pondering these Propositions, in dialogue with the more advanced standpoint of Nicholas of Cusa’s “On Conjectures.” (As noted last week, “The odd number appears to have more of unity than the even number, because the former cannot be divided into equal parts and the latter can be. Therefore, since every number is one out of unity and otherness, so there will be numbers in which the unity prevails over the otherness, and others in which the otherness appears to absorb the unity.”)

From Cusa’s standpoint, the indicated principles of Euclid can be stated as follows: When two even numbers are added, the otherness still prevails over unity, producing an even number. When two odd numbers are added, the unities from each one are combined, making otherness prevail over the unity, and producing an even number. When an even and an odd number are added, the unity of the odd number remains, producing an odd number. In sum when like numbers are added, the otherness prevails over unity, and an even number is produced. When unlike numbers are added, unity prevails over otherness, and an odd number is produced.

When an even number is added an even number of times (i.e., multiplied by an even number), otherness continues to prevail, resulting in an even number. When an odd number is added an even number of times (i.e., multiplied by an even number) otherness prevails and an even number results. When an odd number is added an odd number of times (i.e., multiplied by an odd number), unity still prevails over otherness, and an odd number results.

Different from addition, unlike numbers, when mulitplied, produce even numbers, and like numbers preserve their type. (The reader will find it liberating to demonstrate for yourself, that this is true in all cases; also the reader should discover the similar principle for subtraction and division, and for the second order types of even-even, even-odd, odd-even, and odd-odd, with respect to addition, multiplication, subtraction and division.)

Having discovered so much from the construction of linear numbers, the prisoner extends his experiments into a new domain and now investigates the construction of polygonal numbers. He begins with the polygon with the smallest number of sides, the triangle.

*

* * *

* * * * * *

* * * * * * * * * *

* * * * * * * * * * * * * * *

* @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @

and so on.

Unlike linear numbers, triangular numbers are constructed not by adding one, but by adding the linear numbers themselves. Each successive triangle, contains within it, all previous triangles, plus the next linear number. The added part is called a Gnomon, (denoted in the above figures by the symbol @) which in Greek geometry, means a shape, which when added to a figure, yields a figure similar to the original one. The word Gnomon is derived from the Greek word to know. (The triangular pillar on a sun-dial, which casts the shadow that marks the time, is also called a Gnomon.) In the above representation, each Gnomon is represented by a different symbol.

(The reader is again urged to make your own hand drawings of the construction of triangular numbers, instead of relying on these computer generated representations. Hand drawings are an efficient means of unfolding the cognitive process. When you make these drawings, locate for yourself, the preceeding triangles, in the successive one.)

Triangular numbers are constructed by adding all previous linear numbers together. 1; 1+2; 1+2+3; 1+2+3+4; …; resulting in the series of triangular numbers, 1, 3, 6, 10, …. The differences (intervals) between each triangular number forms the series, 2, 3, 4, 5, …. The difference between the differences is always 1. Here, unity is found, not in the construction of the numbers, but in the differences of the differences.

Intrigued by this discovery, he extends the experiment to the next polygon, the square. Square numbers are constructed thusly.

* @ & \$ # %

* * * * @ @ @ & \$ # %

* * * @ * * * * @ & & & \$ # %

* * @ * * * @ * * * * @ \$ \$ \$ \$ # %

* @ * * @ * * * @ * * * * @ # # # # # %

* @ @ @ @ @ @ @ @ @ @ @ @ @ @ % % % % % %

and so on.

Again, each square contains within it all previous squares, plus the addition of a Gnomon. (The Gnomon with respect to each square is denoted by the symbol @. The last figure represents each Gnomon with a different symbol.) The square numbers increase by adding every second linear number, to the previous square number, 1+3; 1+3+5; 1+3+5+7; resulting in the series of square numbers, 1, 4, 9, 16, 25, 36,… The differences (intervals) between each square number, forms the series of odd numbers, 1, 3, 5, 7, 9, …, and the difference between any two odd numbers is always 2, or is always divisible by 2.

The prisoner can now prove, why these difference are always odd, by looking at the nature of each Gnomon, from the standpoint of his previous discoveries about the nature of even and odd numbers. (When making your hand drawings, distinguish each successive Gnomon and see how each square contains, nested within it, all previous squares. Then look at each Gnomon from the standpoint of the nature of adding even and odd numbers.)

The prisoner now thinks, “Under what conception can I bring the generating principle of the square numbers into a One.” The square numbers are obviously not equal to one another, so equality is not the right conception. But, congruence is not self-evident, as no modulus can be found, relative to which all square numbers are congruent. But the differences (intervals) between the square numbers, (i.e., the odd numbers) while not equal, are all congruent to unity relative to modulus 2. Here the unity is found, not in the formation of the square numbers, nor in the differences between the square numbers, or even in the difference between the differences. Unity is found, as that to which all the differences between the square numbers, are congruent, relative to the modulus of the difference of the differences. (In this case, modulus 2.)

Excited by the ability of his mind to increase its cognitive power, by discovering a congruence, not on the surface, but in the underlying generating principle, he drives the process further. By extending his experiments to polygons of increasing number of sides, the prisoner seeks to force new anomalies to emerge, so he can find what new ordering principles he can discover.

So on to pentagonal numbers. Which he constructs thusly:

*

* *

* * * *

* * * * * *

* * * * * * * * *

* * * * * * * * * * *

* * * * * * * * * * * * *

* * * * * * * * * * * * * *

* * * * * * * * * * * * * * *

and so on.

Here again, each pentagonal number contains nested within it, all previous pentagonal numbers. (Here the reader must make his own hand drawings, as this computer is utterly incapable of doing the work for you, let alone the thinking.) Each pentagonal number increases over the previous pentagonal number by the addition of every third linear number. 1+4; 1+4+7; 1+4+7+10; 1+4+7+10+13; resulting in the series of pentagonal numbers, 1, 5, 12, 22, 35, …. The differences (intervals) between the pentagonal numbers forms the series, 4, 7, 10, 13…. The difference between any two differences is always 3, or is divisible by 3.

Like the square numbers, and the triangular numbers, the pentagonal numbers are not equal, and no modulus can be found, relative to which all pentagonal numbers are congruent. But, when the prisoner looks to the generating principle of pentagonal numbers, a modulus can be found under which the ordering principle can be thought of as a One. The differences between the pentagonal numbers are all congruent to unity relative to modulus 3. Again, unity is found, as that to which all the differences between the pentagonal numbers are congruent, relative to the modulus of the difference of the differences. (In this case, modulus 3.)

This process can be extended to polygons of ever-increasing numbers of sides, forming hexagonal numbers, heptagonal numbers, octagonal numbers and so on. The prisoner spends some time, carefully drawing each series of polygons, so as to bring the generating principle of each polygonal series into a One in his mind. (The reader is well advised to do the same.)

Having done this, a new, more profound question comes before the prisoner’s mind. “What is the generating principle, under which the generating principle of all polygonal numbers can be brought into a One?”

With each new polygon, a new series of numbers is constructed. Unlike linear numbers, which increase by adding one, the polygonal numbers, increase by an increasing amount each time. Each polygonal series, is unified, not with respect to each number of the series, but by the differences between those numbers, which are all congruent to unity relative to a modulus formed by the differences of the differences. (The reader will see that the modulus is always two less than the number of sides of the polygon.)

The prisoner has discovered a generating principle, of a generating principle.

(These discoveries, some of which were embodied in classical Greek knowldedge, were subsequently investigated by Pascal and Fermat, formed a basis for Leibniz’ discovery of the differential calculus, and were reworked by Gauss’ in the development of the complex domain.)

The prisoner steps back and looks at his work, taking a deep breath of fresh air. He feels as though he’s climbed a high peak, on a path whose direction and steepness has changed along the way. The path began with the simple step of adding one, to construct the linear numbers. The path became more curved and the angle of ascent changed, as the concept of numbers was extended into the domain of polygons. Now, at the summit, the change in curvature, and changing angle of ascent, are thought of as a One, under a principle that is congruent with the principle which he started, thought of in an entirely new way. Now the addition of unity, is found, not in the generation of the numbers themselves, but in the generation of the moduli, under which the differences between each polygonal number series are themselves made congruent to unity.

In seeing, with his mind, this whole process from the summit, he asks himself, “What curvature is all this a reflection of?”

His free-thinking is suddenly interrupted by the sound of footsteps. He looks up to see a well-dressed, slightly paunchy baby boomer, with an air of self-importance about him. The man is clutching a very large heavy textbook. As he comes close, the prisoner looks quizzically at the stranger, who sticks out his hand, saying, “Dr. Crumbbucket here. Glad to meet you. I’m a visiting professor, of applied and theoretical bullshit. I understand you’re in need of instruction.”

The prisoner stares for a moment, as the fresh air seems to rush out of his head. His lunch gurgles in his stomach. He prepares to defend his mind.

The Prisoner and the Professor

by Bruce Director

“I have information that you’ve been playing around with numbers,” Dr. Crumbbucket inquired of our prisoner.” Perhaps I could help you to learn the ropes.”

“Well,” our prisoner says slowly, trying to buy some time to collect his thoughts, “I was just sort of making some experiments.”

“Experiments!” Crumbbucket shrieks. “With numbers? No one experiments with numbers. There are well-established rules for the proper manipulations of the figures. Rules which have been handed down from professor to professor, generation to generation. Complicated rules, intricate rules. These take years to learn. Either you can learn these rules, or we give you an electronic calculator with pictures on it. No one can learn by experiments with numbers. There’s nothing to experiment with. Besides, you can’t do experiments in the mind.”

“Not {in} the mind,” the prisoner corrects, “{About} the mind. These experiments are to discover how my own mind thinks.”

“Whatever,” the professor mumbles, after a short pause.

“Do you know all the rules?” The prisoner is still trying to collect his thoughts.

“Virtually all of them. And as soon as a new one is invented, I learn that one, too.”

“Is this what you had to do to get your PhD.?” the prisoner asks.

“Yes. I had to memorize, aggrandize, temporize, fantasize, eulogize, surmise, bastardize, etymologize, generalize, syllogize, tautologize, ventriloquize, analyze, brutalize, formalize, legalize, socialize, symbolize, agonize, fraternize, tyrannize, plagiarize, Anglicize, summarize, and vulgarize, but, I haven’t, at least not yet, had to hypothesize. If you want to learn, we can begin the lessons immediately.”

Crumbbucket’s face is getting redder as he speaks, and small beads of sweat are forming on his forehead and on his chin. The prisoner has a sinking feeling that his whole day is about to be wasted. With no place to go, he has to think fast. Suddenly, a discovery, that, until now was only half-formed in his mind, comes into view. He decides to put the Professor to a test.

“Let me first show you what I’ve discovered by experiment,” the prisoner says.

“Okay, but don’t take long. We have a lot of work to do, if you want to learn what I have learned.”

The prisoner quickly reviews his experiments and discoveries with even, odd and polygonal numbers, to set the professor up for the test.

“That’s no big deal. We have rules for all those things. If you knew the rules, you wouldn’t have had to go through all those manipulations with lines, and dots, and all those drawings. Let’s get on with it.”

“Before we go on, dear Professor, let me put to you a series of questions, so you can better understand the results of my experiments. Are you agreeable to this?”

“If it doesn’t take too long,” the professor answers, shifting his weight from side to side, while one of his knees vibrates quickly back and forth.

“Okay,” the prisoner begins, “Since I’ve already discovered some things about linear and polygonal numbers, I now ask what happens when I add areas?”

“Areas?”

“Yes. Areas. If I have an area whose magnitude is one, and I add another area whose magnitude is one, what is created?”

“Well, that’s obvious. 1 + 1 = 2.”

“And, if I add an area whose magnitude is two to an area whose magnitude is two, what is created?”

“The same: 2 + 2 = 4.”

“And, if I add an area whose magnitude is four and I double it, what happens?”

“The same thing. 4 + 4 = 8. Of course, 2 x 4 = 8 is the same thing. As with 2 + 2 + 2 + 2 = 8. Likewise the same with 2 x 2 x 2 = 8. Or 2^3=8.”

“Okay. Well, let’s try drawing these areas and see what happens?”

“Why do you waste time with drawings?” growled the Professor. “I just showed you how you can add, multiply, or take the powers to get the answer. Why in the devil’s name do you want to waste time with drawings?”

“Just try it. It won’t take long. Here,” the prisoner gently hands the professer his pencil and paper.

“Me? Draw?”

“Yeah, please. Just try it.”

“Whatever,” grumbles the professor, as he reluctantly takes the pencil and paper.

“Now, draw a square whose area is one,” instructs the prisoner. The professor complies, drawing a small square in the middle of the paper. (As usual, the reader is urged to make your own drawings.)

“Now draw another square of the same size, attached to the previous square,” comes the next instruction.

“What has been created?” the prisoner asks.

“A one by two rectangle,” replies the professor.

“And what is the area of the rectangle?”

“Two.”

“See, we’ve added two squares, and we’ve gotten a rectangle,” the prisoner says proudly.

“What’s the difference?” says Crumbbucket dismissively, “I got the same answer following the rules: 1 + 1 = 2. And that was much quicker.”

“Keep going,” says the prisoner, ignoring the professor’s insolence.

“Please, draw another one by two rectangle attached to the one you’ve already drawn. Now what have you created?”

“A two by two square.”

“And what is the area of that square?”

“Four,” the professer responds. “But so what, I already figured the answer. 2 + 2 = 4. Also 2 x 2 = 4.”

“Please. Can we continue?” The prisoner coaxes the professor to continue the drawing. Dr. Crumbbucket draws another four by four square attached to the previous one, making a two by four rectangle whose area is eight. And continuing, drawing another two by four rectangle attached to the previous one, making a four by four square whose area is sixteen.

“See,” the prisoner says excitedly, “First you had a square, and you added a like square, making a rectangle whose area was double the square. Then you added a like rectangle, and you got a square whose area was double the rectangle. Then you added a like rectangle, and you got a square whose area was double the area of that rectangle. As you proceeded, you got another square, then a rectangle. The first addition made a rectangle, the second addition made a square, the third addition made a rectangle, the fourth addition made a square, and so on.”

“But I got the same answer this way, 1×2=2×2=4×2=8×2=16…, or alternatively, 2^0=1, 2^1=2, 2^2=4, 2^3=8, 2^4=16…” answers the professor.

Grinning from ear to ear, the prisoner rejoins, “But from your way, you didn’t discover the series of alternating squares and rectangles. Now you’ve discovered that the odd-numbered additions of areas make rectangles and the even-numbered additions make squares.”

The professor snorts, shrugs his shoulders and says, “Are you ready to learn the rules?”

“Can we try one more series of questions?” asks the prisoner.

“Just one more,” agrees the professor hesitantly, his curiosity getting the better of him.

“Try this,” instructs the prisoner. “Draw a square whose area is the same as the first square, one. Next to that, draw a square whose area is the same as the one by two rectangle, two. And next to that, draw the two by two square, and next to that a square whose area is the same as the two by four rectangle. Do this for all the areas you created by the first series of drawings.”

The professor makes a neat drawing of squares, one next to the other with areas one, two, four, eight, sixteen, and so forth.

“Now, Dr. Professor. What is the length of the side of the first square whose area is one?”

“One, of course,” the professor answers.

“And what is the length of the side of the second square whose area is two?”

“The square root of two,” the professor states matter of factly.

“And what is the square root of two?”

“It’s the length of the side of the square whose area is two, and is denoted with a symbol thusly,” the professor responds without blinking, tracing a radical sign in the air with his finger.

“But,” replies the prisoner, “I already know the area of the square is two. You are simply repeating yourself, to tell me that the length of the side, is `The length of the side of the square whose area is two.'”

“The square root of two,” the professor repeats, more emphatically than before.

“But that doesn’t say anything. What’s the square root of two?” the prisoner asks again. “Can we continue? What is the length of the side of the next square, the one whose area is four?”

“Two,” answers the professor.

“Very fine. And what is the length of the side of the next square whose area is eight?” asks the prisoner.

“The square root of eight.” This time the professor’s pride in his ability to answer is tinged with trepidation, anticipating the prisoner’s response.

“There you go again. You have only repeated the question as the answer. I ask, `What is the length of the side of a square whose area is eight?’ and you answer, `The length of the side whose area is eight.’ That is not an answer. From that, we have discovered nothing.”

Perceiving the professor’s obvious distress, the prisoner tries to be gentle, hoping that his prodding will liberate the professor’s mind.

The professor stares for a moment in disbelief at the resistance of the prisoner to accept his answer.

The prisoner asks again, “What is the length of the side of the square whose area is two or eight? Or in your words, what is the square root of two, or eight?”

“Here, hold this,” the professor hands back the pencil and paper after the briefest moment’s pause, and picks up his heavy textbook, wildly flipping the pages. “I know it’s in here somewhere,” he says as he balances the book in one hand, turning the pages with the other. The prisoner stands mute with a wry smile on his face.

“Just a minute. I’ll find it,” begs the professor. “Damn it! Wrong book. Hang on a minute. Don’t go away, I’ll be right back. I have to get my other book.”

“I shall return,” the professor calls, his voice trailing off. The prisoner watches the professor scurry down the hall, half of his shirt-tail hanging out of his pants, the sound of clanging chains diminishing as he gets further away.

Free from the immediate encounter with the professor, the prisoner turns his thoughts back to the drawings just created. He spends some time making similar drawings, in which he increases the area by three each time, then by four, then by five. Each time he creates an alternating series of squares and rectangles, with the first addition being a rectangle, the second a square, the third a rectangle, and so forth. The rate at which the areas grow, changes, but the type of change in each case is the same; the odd-numbered additions (powers) make rectangles, the even-numbered additions (powers) make squares. He has discovered even and odd, in a new domain.

When he added linear numbers, thus forming polygonal numbers, the rate of growth changed for each type of number, but remained the same within each series. Among linear numbers, he discovered congruences, between even and odd, between even/even, even/odd, odd/even, and odd/odd. Among the polygonal numbers, he discovered congruences with respect to the change between each number. Areas (geometric numbers), reflect an entirely different type of change, as the numbers are increased.

These two-dimensional geometric numbers, reflect a new domain. Congruence with respect to even and odd remains, but in an entirely different way. Here, evenness reflects squares and oddness reflects rectangles. When these magnitudes are transformed into only squares, the sides of the even ones are commensurable with the area, while the sides of the odd ones are incommensurable with the area. The concept of number cannot be seperated from the content of number, which is a reflection of the domain in which that number is situated. Even something as seemingly simple as even and odd, is different in different domains. The poor professor didn’t even suspect, that from the method he used, he really didn’t know the area of half the squares he drew, even though he seemed to be able to draw them. “There’s probably some hope for him,” the prisoner thinks to himself, “if he’ll only try to discover, rather than just learn.”

The prisoner asks himself again, “What curvature do these processes reflect?” In his mind’s eye, he sees, with respect to each type, different principles of growth, which are reflected as a series of nested curves; a circle, an Archimedean spiral, and an equiangular spiral. Each type of curvature, is reflected simultaneously, yet distinctly, in each process.

Now he thinks of a new, most important project: “What is the nature of the curvature, which bounds these curves?”

## The Circle Is Not Simply Round

by Bruce Director

Among the most interesting and provocative investigations of the thinkers of the ancient Greek speaking world, were problems concerning the construction, with straight edge and compass, of certain geometrical figures; specifically, the doubling of the cube, the trisection of the angle, the construction of the regular heptagon, and the quadrature of the circle. In most of the modern English language sources on the subject, these problems are generally portrayed as a certain type of puzzle, or brain teaser. Lacking in virtually all of this scholarship, is any conception of what these ancient Greek scientists were actually investigating. To answer the latter question, we need not hunt for some long-lost text, in which the deeper implications of these investigations are explicated. Rather, we need only to relive the discoveries ourselves, and, in the mirror of our own mind, those deeper implications will be reflected.

Instead of wasting time with today’s academics, let us take as our guide Johannes Kepler, whose new and original discoveries arose from his own re-working of these investigations of ancient Greece. In the first book of the Harmony of the World, “The Construction of Regular Figures,” Kepler presents some of the results of his re-discovery. Pertinent to this discussion, he provides the following definitions:

VII. “In geometrical matters, to know is to measure by a known measure, which known measure in our present concern, the inscription of figures in a circle, is the diameter of the circle.”

VIII. “A quantity is said to be knowable if it is either itself immediately measurable by the diameter, if it is a line; or by its [the diameter’s] square if a surface: or the quantity in question is at least formed from quantities such that by some definite geometrical connection, in some series [of operations] however long, they at last depend upon the diameter or its square. The Greek word for this is `gnorimon.’

IX. “The construction of a quantity which is either to be described or to be known is its deduction from the diameter, by permitted means, in Greek [these are called] `porima.’

“So construction generally yields either description or knowledge. But description declares mere quantity, whereas knowledge also in addition declares quality or a definite quantity. Now a line can be geometrically determined, in Greek, `takah,’ even though its quality is not yet known intellectually. On the other hand, a line or lines may be known qualitatively, but that does not yet determine them or make them determinate, that is to say if their quality is common to many other things which are different in quantity. So for such lines description is easy, knowledge very difficult. Finally, many things can be described by some Geometrical means or other; but cannot be knowable by their nature: as knowledge has been defined above.”

With these concepts in mind, take a first look at one of the classical Greek problems, the trisection of the angle. In proposition #46 of the same book Kepler restates this problem as:

“The division of any arc of a circle into three, five, seven, and so on, equal parts, and in any ratio which is not obtainable by repeated doubling from the ones which have been shown above, cannot be carried out in a Geometrical manner which produces knowledge.”

His demonstration goes like this: To bisect an arc of a circle, we first bisect the chord drawn between the two ends of the arc. A line drawn from the center of the circle, through that point, will also bisect the arc. On the other hand, if we want to trisect the arc, the situation becomes much more ambiguous. Draw an arc and its chord, and label the end points A and B. Construct points P and Q on the chord, such that A-P = P-Q = Q-B. This trisects the chord. If we now draw lines from the center of the circle through P and Q, that intersect the arc at P’ and Q’, the arcs A-P’ and Q’-B will be smaller than the middle arc P’-Q’. On the other hand, if we draw lines perpendicular to the chord, through P and Q, the result will be that arc P’-Q’ will be larger than the arcs A-P’ and Q’-B. Therefore, to trisect the arc, we have to draw lines through P and Q from a point that is somewhere between the center of the circle and infinity. Kepler shows that the position of this point gets farther from the center, as the arc gets smaller. But, this relationship is not proportional. That is, decreasing the arc by a given amount, does not change the position of the point by a proportional distance.

This is another manifestation of the phenomenon of non- linearity of circular action demonstrated three weeks ago with respect to the sine and cosine. One can illustrate that principle with the following experiment:

Draw a large circle on a black board. Draw two perpendicular diameters, one vertical and one horizontal. Get a string with a weight on it. Hold one end the string with your finger at the intersection of the horizontal diameter and the circumference, and let the weight hang down towards the floor. Now, move the end of the string with your finger along the circumference of the circle. The weight will rise, and the string will form a chord of the circle that intersects the horizontal diameter. Now watch the intersection of the string and the diameter, as you move the end of the string around the circumference of the circle. What is the relationship between the circular action of your finger, and the rectilinear movement of the point of intersection of the string with the diameter? The constant curvature of the circle, produces a non-constant motion of this point.

Back to the trisection of the angle. What Kepler’s demonstration reveals, is a kind of boundary condition with respect to the divisions of a circular arc. Dividing a circular arc in half, or into powers of two, does not produce an immediate discontinuity between the circular arc and the straight line chord defined by it. But when we try to divide by three, division of the line and the arc diverge.

Through Kepler we have now re-discovered this problem in the form confronted in 5th-4th Century B.C. One attempted solution was devised by Hippias of Elis<fn1>, who is credited with producing the first non-circular curve, today called the quadratrix. This curve was later investigated by Leibniz, Huygens, Bernoulli, et al., from the higher standpoint of that Leibniz developed out of Kepler’s discoveries.

The quadratrix of Hippias is generated as follows. Draw a square. Label the corners clockwise from the upper left hand corner A, B, C, D. Now imagine side A-D rotating clockwise around point D. As this segment rotates, point A will trace a quarter of a circle from A to C. Now imagine that as this line is rotating, side A-B, moves down the square to side D-C, at the same rate as side A-D is rotating. Side A-B remains parallel to side D-C as it moves. The quadratrix is the curve traced out by the intersection of these two lines as they move. Thus, the quadratrix is the intersection of circular rotation and linear motion.

Because of the way the quadratrix is constructed, Hippias used it to trisect the angle. That is, since both sides move at the same rate, when A-D has rotated 1/2 way, side A-B has moved down 1/2 way. Similarly, when A-D has rotated 1/3 of the way, side A-B has moved 1/3 of the way down. And so on for any other division.

To trisect an angle, mark off any angle with vertex at D, such that one side of the angle will be D-C, and the other side of the angle will intersect the quadratrix at some point K and the arc A-C at L. Draw a line parallel to the side D-C through K. That line will intersect side A-D at some point E. Now, find the point E’ that divides D-E by one third. Draw a line from E’ that is parallel to side D-C. This line will intersect the quadratrix at some point K’. Connect K’ to D and the angle formed with side D-C will be one third of the original angle.

But, has Hippias constructed a means for trisection that is a “knowable” quantity as Kepler re-stated the problem? We could construct an mechanical apparatus to draw a quadratrix, but is this curve “knowable,” that is, constructable by the circle and it’s diameter?

Ah, there’s the rub! To construct the quadratrix by “knowable” means, Hippias proposed the following:

First draw the perpendicular bi-sector of side D-C. Then draw the bisector of the angle C-D-A. The intersection of these two lines is a point on the quadratrix. Then bisect the two new segments of side D-C and the two new angles formed by bisecting C-D-A. These intersections will define two more points on the quadratrix. This process can be repeated again and again, to fill in so many points on the quadratrix, that by connecting the dots, the quadratrix can be drawn.

But, wait! All these points were determined by division by two, and so they will only precisely determine points on the quadratrix that intersect lines that divide angle C-D-A by powers of two. None of these dots will precisely determine a point on the quadratrix that trisects an angle. Those points, will always lie on the indeterminate, “filled” in parts of the curve. The “non-knowable” parts.

We seem to have hit a stumbling block. The boundary between division of a circular arc by two and division by three has re- emerged. Think about this a while. The closed door we’ve seem to run into, is, perhaps, an open hallway, through which our predecessors have strolled.

1. This Hippias is the subject of Plato’s dialogues Hippias Major and Hippias Minor. He is also mentioned in the Apology. He was apparently a traveling philosopher, with some facility in mathematics.

Double Your Mind –

by Bruce Director

This week we look at another classical Greek problem as reported by Theon of Smyrna:

In his work entitled {Platonicus} Eratosthenes says that, when the god announced to the Delians by oracle that to get rid of a plague they must construct an altar double of the existing one, their craftsmen fell into great perplexity in trying to find how a solid could be made double of another solid, and they went to ask Plato about it. He told them that the god had given this oracle, not because he wanted an altar of double the size, but because he wished, in setting this task before them, to reproach the Greeks for their neglect of mathematics and their contempt for geometry.

In future weeks we will re-live this problem. For now think about it.

The Means to Double Your Mind

by Bruce Director

According to Eratosthenes, Plato took great pleasure at the prospect, that the cognitive capacity of his fellow Greeks might be improved, when, having asked the gods for help, the gods answered with a question, that required the Greeks to think. It might improve our own cognitive functioning, and make us smile, to find out why Plato was so delighted.

Before taking up this problem directly, let’s first look at the Pythagorean investigation of the doubling of the square. Plato discussed these investigations in the famous Meno dialogue. We can reconstruct an essential feature of that discovery by the following means:

Draw a square. Double its side and draw the square on the doubled side. Repeat this process several times. If the area of the first square is considered 1, then the subsequent areas are 1, 4, 16, 32 … etc. The sides of the corresponding squares are 1, 2, 4, 8, 16, …

Now draw another square. Draw the diagonal. Draw a square on the diagonal. Draw the diagonal of the new square. Draw a third square on the diagonal of the second square. Continue this several more times.

You should see a series of squares and diagonals in a spiral formation. If the area of the first square is considered to be 1, then the subsequent squares have the areas, 1, 2, 4, 8 …, respectively. The area of each square in this series is in the same proportion to the one proceeding it, as to the one succeeding it. That is, 1:2::2:4::4:8::8:16 …

Notice that the squares produced by doubling the sides are every other one, of the series of squares produced from the diagonals. The squares of doubled area, are in between the squares of doubled sides. If we think of the curvature of the whole series, the squares of double areas are found in a smaller interval of action, than the squares of doubled sides. The proportionality between the terms of the series remain the same, but something completely new emerges in the smaller interval of action, to wit: doubled areas.

(It should be particularly thought provoking, to think of the curvature in the small, of a discontinuous series!)

Now look at an even smaller interval of action; the interval between two squares. That interval includes, the side of one of the squares, the diagonal, and the side of the succeeding square. Here again, the principle of proportionality remains the same, but a new singularity emerges — the incommensurability of the side to the diagonal of the square. (For a more complete discussion of incommensurability, see “Incommensurability and Analysis Situs” Parts 1 and 2; NF Vol. XI #22 6/9/97 97237jbt101 and NF Vol. XI #23 6/16/97 (7267jbt101).

In Plato’s Theatetus dialogue, Theatetus reports on an investigation of a still smaller interval of action of this same curvature, by Theodorus of Cyrene, a Pythagorean who was one of Plato’s teachers. Theodorus produced a series of triangles whose hypotenuses were the square roots of 2, 3, 4, 5, etc. For now, we leave to the reader the fun of re-constructing this series.

Now think back on these series of squares. All are characterized by the same principle of proportionality. Each one is a smaller interval of action than the previous one. With each smaller interval, new singularities emerge. What is invariant in every smaller interval, however, is the principle of proportionality.

Now if a square is doubled through this principle of proportionality, how is the cube doubled? A generation or two before Plato, Hippocrates of Chios, (not the same Hippocrates of medicine fame) made a crucial discovery concerning this problem. This Hippocrates supposedly was a merchant who ended up broke in Athens around 500 B.C. and started to teach thinking to earn a living. His discovery can be re-created in the following way:

If we begin with a cube whose side is one, its volume will also be one. If we double the side, the new cube will have a volume of 8. Between 1 and 8 are two means, 2 and 4. That is, 1:2::2:4::4:8. So, the cube whose volume is double, is the lesser of the two means, between the cube whose side is doubled. Since the sides of the cubes are in the same proportion as the volumes, the side of the cube whose volume is double, is the lesser of two means between 1 and 2.

Eutocius reported this discovery in his commentaries on Archimedes as, “It became a subject of inquiry among geometers in what manner one might double the given solid, while it remained the same shape, and this problem was called the duplication of the cube; for, given a cube, they sought to double it. When all were for a long time at a loss, Hippocrates of Chios first conceived that, if two mean proportionals could be found in continued proportion between two straight lines, of which the greater was double the lesser, the cube would be doubled, so that the puzzle was by him turned into no less of a puzzle.”

Plato reflected on this discovery in the Timeaus: “But it is not possible that two things alone should be conjoined without a third, for there must needs be some intermediary bond to connect the two. And the fairest of bonds is that which most perfectly unites into one both itself and the things which it binds together; and to effect this in the fairest manner is the natural property of proportion. For whenever the middle term of any three numbers, cubic or square, is such that the first term is to it, so is it to the last term, and again, conversely, as the last term is to the middle, so is the middle to the first,– then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity. Now if the body of the All had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow-terms, but now it is otherwise, for it behoved it to be solid of shape, and what brings solids into unison is never one middle term alone but always two.”

Hippocrates’ discovery amounts to investigating the geometric principle of proportionality, in an even smaller interval, trying to find two means between 1 and 2. His successors soon discovered a new boundary had to be crossed, when investigating this smaller interval. Those familiar with Gauss’ Disquisitiones Arithmeticae and his theory of bi-quadratic residues, will recognize the seeds of those great investigations, in these ancient Greek inquiries.

In future weeks we will re-construct some of the ancient Greek studies of this problem. For now, keep thinking about it.

## Beyond Counting — A Preparatory Experiment

by Bruce Director

In previous pedagogical discussions on Higher Arithmetic, we investigated the ordering of numbers with respect to arithmetic (rectilinear) progressions, (as in the case of linear and polygonal numbers) and geometric (rotational) progressions, as in the case of geometric numbers, and prime numbers. (See Doc.#’s 97267bmd01; 97316bmd001; 97326bmd001;)

The deeper implications of these investigations, which form the basis of Gauss’ re-working of Greek classical geometry, reveal themselves, only if we rise above intuition, and investigate the nature of numbers with the mind only.

In the coming weeks, we will begin to further investigate these principles. But, it will be much more efficient, if the reader first performs the following experiments:

As discovered in an earlier pedagogical discussion, the geometric progression is constructed by beginning with a square, whose sides are a unit length, and whose area is a unit area. We then add 2, 3, 4, or more squares forming a rectangle. We then double, triple, quadruple, this rectangle forming a new square, and so forth. With each successive action, the area of the corresponding square or rectangle increases, but the type of action, doubling, tripling, quadrupling, etc. doesn’t change. A different type of number, incommensurable with rectilinear numbers, is discovered by this process. (This construction is discussed by Plato in the beginning of the Theatetus dialogue.)

In contradistinction, the rectilinear (polygonal) numbers, form a series in which each number associated with a given polygon, increases by an increasing amount, but the differences between the differences remains the same. And so, under Gauss’ concept of congruence, all polygons of the same type can be brought into a One, because the differences are all congruent, relative to a modulus which is the number of sides minus 2. The totality of all polygons, can be thought of as a series of series, ordered by successively increasing moduli.

For geometric numbers, however, there is no simple modulus, under which the individual members of any given geometric progression can be made congruent. Or, put another way, the change from rectilinear (1 dimension) to rotational (2 dimension) changes the ordering principle. We must shift tactics. The old rules, don’t apply. We must discover a new, higher type of congruence. This new higher type of congruence, opens the door to whole new domain.

To discover the nature of this domain, it is most efficient to follow in Gauss’ footsteps, and first discover these orderings experimentally, and then investigate the deeper implications, which underlie these orderings.

Each different geometric progression can by also thought of as a series of numbers associated with the underlying action, in order of increasing actions. For example, 2 for doubling. The first number in the series is a unit area, which has undergone no doubling, i.e., 2^0 or 1. The second number is the first doubling, or 2^1 or 2. The third number is the second doubling or 2^2 or 4. The third number is the third doubling, or 2^3 or 8, etc. This forms the geometric progression, 1, 2, 4, 8, 16, ….

Another example, 3 for tripling. The first number is the unit area which has undergone no tripling or, 3^0 or 1. Then the first tripling, 3^1 or 3; the second tripling 3^2 or 9; the third tripling 3^3 or 27. This forms the series 1, 3, 9, 27, ….

Now investigate the congruences of these series with respect to odd prime numbers as moduli. Begin with modulus 3. Calculate the least positive residues of the numbers of the geometric progression based on 2 with respect to 3 as a modulus. Then take 5 as a modulus. Calculate the least positive residues of the numbers of the geometric progressions based on 2, 3, 4, with respect to 5 as a modulus. Then take 7 as a modulus. Calculate the least positive residues of the numbers of the geometric progressions based on 2, 3, 4, 5, 6 with respect to modulus 7.

What new type of orderings emerge? What’s going on here? We will begin to investigate these questions, next week.

Beyond Counting — Part II

by Bruce Director

If you carried out the experiment in last week’s discussion, you would have discovered the reflection of an ordering principle with respect to the residues of geometric progression. The experiment should have yielded the following result.

With respect to modulus 5, the residues of the geometric progressions based on the numbers 2-5 yield the following results: (The Powers are in the first row; the residues resulting from a specific geometric progression are in the rows which follow. The base is the type of action from which the geometric progression is generated — 2 for doubling; 3 for tripling; etc.).

```Powers: 0 1 2 3 4 5 6 7 8 9 10
Base 2: 1 2 4 3 1 2 4 3 1 2 4 etc.
Base 3: 1 3 4 2 1 3 4 2 1 3 4 etc.
Base 4: 1 4 1 4 1 4 1 4 1 4 1 etc.```
```For Modulus 7:
Powers: 0 1 2 3 4 5 6 7 8 9 10
Base 2: 1 2 4 1 2 4 1 2 4 1 2 etc.
Base 3: 1 3 2 6 4 5 1 3 2 6 4 etc.
Base 4: 1 4 2 1 4 2 1 4 2 1 4 etc.
Base 5: 1 5 4 6 2 3 1 5 4 6 2 etc.
Base 6: 1 6 1 6 1 6 1 6 1 6 1 etc.```

This is a surprising result. The unbounded, ever increasing geometric progression, is brought into a simple periodic ordering with respect a prime number modulus. No matter which type of change (base) of the geometric progression, a periodic cycle emerges with respect to a prime number modulus. Each period, begins with unity, making a sort of wave pattern. While the “wavelength” may change with the base, the “wavelength” is always either the modulus minus 1 (m-1) or a factor of m-1. No other “wavelengths” are possible. The bases whose “wavelengths” are m-1 are called “primitive roots.” (In the examples above, 2 and 3 are primitive roots of 5; 3 and 5 are primitive roots of 7.)

These orderings were investigated by Fermat and Leibniz, and, according to Gauss, Leibniz’ investigations of these orderings, were a subject of the oligarchical slave Euler’s attack on Leibniz, played out in the famous fight between Koenig and Maupertuis. In his Disquisitiones Arithmeticae and the two Treatises on Biquadratic residues, Gauss unfolds even deeper implications of these orderings, which will be discussed in future pedagogical discussions. For now, it is sufficient to reflect on the subjective questions presented by the phenomena.

In order to even begin to discover what’s going on here, you must think in an entirely different way about numbers. What accounts for these orderings? The answer will elude you, if you cannot free yourself from a conception of number associated with mere quantity of objects. Just as the discovery of valid physical principles, such as the orbit of the asteroid Ceres, will elude you, if you cannot free your mind from fixating on the mere observations. The answer lies outside the orderings themselves, and can only be reconstructed inside the mind, by reflecting on the paradoxes presented.

Instead of thinking of each number individually, think instead of a series from 1 to m-1, associated with a unique principle of generation, that contains each number. Each principle of generation is characterized by a distinct type of curvature. One principle of generation, is the principle of adding one (rectilinear). Another principle of generation is the principle of adding areas (sprial action). A third principle of generation, is the principle of congruence (circular rotation). A fourth principle of generation is the principle of prime numbers. The combination of all four characterizes a hypergeometry, the unfolding of which, generates the periodic orderings reflected in the residues of powers.

The subjective challenge, is to be able to conceive in your mind of the interconnection of these generating principles as a One, when that One cannot be expressed as a mathematical function. The functional relationship exists only in the mind. Just as the One of a musical composition exists not in the notes, or the physical characteristics of the well-tempered system, but in the Idea of the composition, which is transfinite with respect to the unfolding of the composition.

In the interest of not diverting attention from concentrating on these subjective questions, the reader is advised to continue these experiments with respect to the prime numbers 11, 13, 17, and 19. In future discussions, we will rediscover Gauss’ application of this principle in his re-working and superseding classical geometry.

## Archimedes and The Student

Bruce Director

To Archimedes came a youth desirous of knowledge.

“Tutor me,” spake he to him, “in the most godly of arts,
Which such glorious fruit to the land of our father hath yielded
And the walls of the town from the Sambuca preserv’d!”
“Godly nam’st thou the art?” She is’t, “responded the wise one;
“But she was that, my dear son, ere she the state ever serv’d.
Wouldst thou but fruits from her, there too can the mortal engender;
What the Goddess doth woo, seek no the woman in her.”

This poem by Friedrich Schiller (translated here by Will Wertz) was cited by Carl F. Gauss in his famous introductory lecture on astronomy. In attacking the pragmatic thinking and the “indifference and insensibility to the great and that which honors humanity,” Gauss told the audience of faculty and students at Goettingen University, the real life implications of this way of thinking. “Unfortunately one cannot conceal the fact that one finds such a mode of thinking very prevalent in our age, and it is probably quite certain that this attitude is very closely connected with the ill fortune which of late has struck so many states. Understand me correctly, I am not speaking of the very frequent lack of feeling for the sciences themselves, but of the source from which this flows, of the tendency everywhere to ask first about the advantage and to relate everything to physical well-being, of the indifference to great ideas, of the aversion to effort due merely to pure enthusiasm for the thing in itself. I mean that such characteristics if they are predominating, can have given a strong decision in the catastrophes which we have experienced.”

He then harkened back to one of his Greek predecessors, “The great happy minds who have created and extended astronomy as well as the other more beautiful parts of mathematics, were certainly not fired by the prospect of future utility; they sought truth for its own sake and found in the success of their efforts alone their reward and their good fortune. I cannot avoid reminding you here of Archimedes, who was admired most by his contemporaries only on account of his ingenious machines, on account of their apparent magic effects; he however valued all this so slightly in comparison with his glorious discoveries in the field of pure mathematics, which at that time mostly had no visible utility in themselves according to the usual sense of the word, that he wrote nothing about the former for posterity, while he lovingly developed the latter in his immortal works. You certainly all know the beautiful poem by Schiller, {Archimedes and the Student}….

Is Gauss asking you to choose between “pure mathematics” and pragmatism? If Gauss’ words seem a bit politically incorrect to you, you have succeeded in identifying a pragmatic demon in your own mind.

Take a look at Archimedes correspondence with Eratosthenes, entitled, “The Method of Treating Mechanical Problems.” In that work, Archimedes poses the problem of determining the relationship between incommensurable solids, such as a sphere, cone and cylinder, or a spheroid and a cylinder. Gauss would later identify that these solids are characterized by different curvatures. The cone and cylinder are generated by surfaces of zero curvature; the sphere by a surface of constant curvature; the spheroid by non-constant curvature.

The problem Archimedes posed to Eratosthenes was: “Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer [of mathematical inquiry], I thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge….”

An example of this method is the determination that the volume of a sphere is four times the cone with base equal to a great circle of the sphere and height equal to its radius; and the volume of a cylinder with base equal to the great circle of the sphere and height equal to the diameter is 1.5 times the sphere.

(The second part of this proposition, was depicted on Archimedes tombstone.)

The difficulty in proving this proposition, lay in the differing curvatures of the volumes measured. To overcome this obstacle, Archimedes investigated the interaction of these volumes in a physical process, the pull of the Earth’s gravity.

To construct the experiment, think of the sphere cone and cylinder, all nested together in the following way. Think of a sphere, then, think of a cone whose base is formed by the equator of the sphere, and whose apex is at the north pole. Now think of that whole thing embedded in a cylinder whose bases touch the north and south pole of the sphere. The sphere will be tangent to the cylinder at its equator.

(You can draw a cross section of this arrangement, by drawing a circle with two perpendicular diameters. This represents a cross section of the sphere. Label the intersections of these diameters counter-clockwise A,B,C,D. Then draw a triangle with vertices A,B,D. This represents a cross section of the cone. Then draw a square around the circle such that A,B,C,D intersect the midpoints of the sides of the square. This represents the cross-section of the cylinder. Label the corners of the square, V,X,W,Y clock-wise from the upper left. And label the center of the sphere K.)

Now, extend the sides of the cone (A-B and A-D in the cross section drawing) and the bottom side of the square (W-Y in the drawing). Label the intersections, E and F. Now we can think of an enlarged cone, A, E, F. Also construct, an enlarged cylinder, with base E-F and height, A-C.

Then, Archimedes imagines that this entire complex of solids is resting on a balance, whose bar is twice the length of A-C, with A at the midpoint. (To depict this in our cross-section drawing, extend double line A-C and label the new endpoint H.)

Finally, draw a line perpendicular to C-A-H and parallel to B-D. This line will intersect the cross sections of all the figures previously imagined. Archimedes, using the Pythagorean theorem, and the principles of Euclidean geometry, determines the proportional relationships existing among these cross-sections.

To determine the relationship of the volumes of the sphere, cone and cylinder, Archimedes investigates under what conditions the various cross-sections of the cone, cylinder and sphere are balanced. Using the proportions he just calculated, he is then able to determine that the volume of the sphere is 4 times the cone and the volume of the cylinder is 1.5 times the sphere.

(The actual calculation is not difficult, but it would be too cumbersome to describe in this format. If you don’t have a copy of Archimedes piece, send an e-mail to BMD, and I will supply a copy of this proposition.)

Is Archimedes procedure a physical demonstration, or a mathematical one. Isn’t he investigating geometrical objects, which have no physical existence, with respect to a physical process, the pull of Earth’s gravity? As he said to Eratosthenes, this procedure makes the proposition clear, but he still requires a geometrical proof, or, as Kepler would later state with respect to the divisions of the circle, “knowability”?

Forty years after Gauss identified the consequences of pragmatism on the political condition of Europe, then former U.S. President John Quincy Adams gave a speech in Cincinnati, Ohio on the occasion of the laying of the cornerstone of the U.S.’s first astronomical observatory. After an extensive discussion of the history of astronomy, Adams ended his speech:

“But when our fathers abjured the name of Britons, and `assumed among the powers of the earth, the separate and equal station, to which the laws of Nature, and of Natures, God entitles them,’ they tacitly contracted the engagement for themselves, and above all, for their posterity, to contribute, in their corporate and national capacity, their full share; aye, and more than their full share, of the virtues, that elevate, and of the graces that adorn the character of civilized man….

“… We have been sensible of our obligation to maintain the character of a civilized, intellectual, and spirited nation. We have been, perhaps, over boastful of our freedom, and over sensitive to the censure of our neighbors. The arts and sciences, which we have pursued with most intense interest, and persevering energy, have been those most adapted to our own condition. We have explored the seas, and fathomed the depths of the ocean, and we have fertilized the face of the land. We–you- -you, have converted the wilderness into a garden, and opened a paradise upon the wild. But have not the labors of our hands, and the aspirations of our hearts, been so absorbed in toils upon this terraqueous globe, as to overlook its indissoluble connection even physical, with the firmament above? Have we been of that family of the wise man, who, when asked where his country lies, points like Anaxagoras, with his finger to the heavens.

“Suffer me to leave these questions unanswered. For, however chargeable we may have been, with inattention or indifference, to the science of Astronomy, heretofore — you, fellow citizens, of Cincinnati — you, members of the Astronomical Society, of this spontaneous city of the West, will wipe that reproach upon us, away….”

## Prime Numbers

By Bruce Director

“Can we deny that a warrior should have a knowledge of arithmetic?…

“…. It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul towards being…

“…. For, if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul, perplexed and wanting to arrive at a decision, asks, `Where is absolute unity?’ This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.

“And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?

“Certainly.

“And all arithmetic and calculation have to do with number?

“Yes.

“And they appear to lead the mind towards truth?

“Yes, in a very remarkable manner.

“Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician….

“… Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study {until they see the nature of numbers with the mind only;} nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being….”

–Plato’s {Republic} Book VII

Elementary considerations concerning prime numbers, directly present us with the fundamental questions to which Socrates refers in the above passage. To grasp this, however, one must confront, and overcome, any influence of Euler and the Enlightenment, in one’s own thinking.

What are Prime Numbers?

Prime Numbers are integers, which are indivisible, by any other number, except one and itself. Composite numbers, are integers, which can be divided by another number. In the {Elements}, Euclid presents a proof that any composite number is divisible by some prime number. Since, any composite number by definition can be divided by another number, that other number is either another composite number or a prime number. If it is a prime number, the case is proved. If it is a composite number, that new number can be divided by another number, which is either a prime number or a composite number. By this method, you will eventually get to a prime number.

A method for discovering which integers are prime, was developed by Eratosthenes in approximately 200 B.C., known as the Sieve of Eratosthenes. This is a method for eliminating the composite numbers, from a group of integers, leaving only the primes.

List the integers from 1 to any arbitrary other integer A. Now, beginning with 2 (the first prime number after 1), strike from the list, all numbers divisible by 2, for they are composite numbers. Do the same with 3 (the next prime number), then 5, etc., until you come to the first or prime is first.

This raises the question, what happens when you try to construct all integers from the primes alone? First, you’d make all the integers composed only of 2. Then you’d make all the integers composed only of 3 and combinations of 2 and 3, and so forth with 5, etc. As you can see, this process would eventually generate all the integers, but in a non-linear way. Compare that process with constructing the integers by addition? Addition generates all the integers sequentially, by adding 1, but does not distinguish between prime numbers and composite numbers.

The unit 1 is indivisible, with respect to addition. With respect to division, the prime numbers are indivisible. Both processes will compose all the integers, but that result coincides only in the infinite. In the finite, they never coincide. The difference, is between the mental act of addition, and the mental act of division. Don’t try resolve the matter, by asking if division is superior to addition. Instead, reflect on that which is different between the two processes, the “in-betweeness.” It is the relations between the numbers, which is the object of our thought, not the numbers in themselves.

This anomaly, is a reflection of the truth, that there exists a higher hypothesis which underlies the foundations of integers. An hypothesis, which is undiscoverable if limited to the domain of simple linear addition. By reflecting on this anomaly, we begin, as Socrates says, “to see the nature of number in our minds only.” Our minds ascend, as Socrates indicates, to contemplate the nature of true being. We ask, “If the domain of primes is that from which the integers are made, what is the nature of the domain, from which the primes are made?”

View the above result from the standpoint of Leibniz’s {Monadology}:

“29. Knowledge of necessary and eternal truths, however, distinguishes us from mere animals and grants us {reason} and the sciences, elevating us to the knowledge of ourselves and of God. This possession is what is called our reasonable soul or {spirit}.

“30. By this knowledge of necessary truths and by the abstractions made possible through them, we also are raised to {acts of reflection} which enable us to think of the so-called {self} and to consider this or that to be in us. Thinking thus about ourselves, we think of being, substance, the simple and the composite, the immaterial, and even of God, conceiving what is limited in us as without limit in him. These acts of reflection furnish the principal objects of our reasoning.” [bmd]

Prime Numbers, Part II

“The natural, sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality. So much, indeed, is number shown to be the beginning of those things which are attained by rationality, that with its sublation, nothing remains at all, as is proven by rationality. And if rationality unfolds number and employs it in constituting conjectures, that is not other than if rationality employs itself and forms everything in its highest natural similitude, just as God, as infinite mind, in His coeternal Word imparts being to things. There cannot be anything prior to number, for everything other affirms that it necessarily existed from it….

“The essence of number is therefore the prime exemplar of the mind. For indeed, one finds impressed in it from the first trinity or the unitrinity, contracted in plurality. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

–Nicolaus of Cusa, “On Conjectures”

Last week, as Socrates says, we, began “to see the nature of number with the mind only.” This week, we further unfold the essence of number, requiring our mind, to lift itself into a new higher domain. This journey may be difficult, at points, for the reader. For those with previous mathematical training, infected by the Euler-Lagrange-Cauchy fraud, you will find your previous training an annoying distraction. For those without such annoying distractions, you may find the lack of such, an annoying distraction in itself. To avoid these distractions, follow the proscription of Plato, Cusa and Leibniz, and see with your mind only. For this purpose, we continue our investigation using the principles of higher arithmetic, as developed by C.F. Gauss, which considers only the relations between whole numbers. Gauss, elaborated a visualization of the complex domain, translated by Jonathan Tennenbaum, in the Spring 1990 issue of Twenty First Century, under the title “Metaphysics of Complex Numbers,” to which the reader is referred.

In the previous discussion, we discovered an hypothesis underlying the essence of number, through reflection on the prime numbers. By its extension, we will be lead to a new, higher hypothesis underlying the essence of number.

First, extend the idea of number, from positive whole numbers, to include their opposites, the negative whole numbers. Here, prime numbers, maintain their same relationship with respect to all numbers, with the exception of a change of direction from positive to negative. Whereas, positive whole numbers are formed sequentially by adding one, negative whole numbers are formed sequentially by subtracting one, or, adding -1. Positive composite numbers are formed by muliplying the prime factors, negative composite numbers are formed by multiplying the prime factors by -1. Think of positive and negative, not as position, but as directions, opposite one another. If positive is right, negative is left. If positive is up, negative is down. One dimension–two directions.

In short we have extended our concept of number, by conceiving of a two-fold unity: 1 and -1.

Once our concept of number is extended into the negative direction, an anomaly immediately appears. All positive and negative whole numbers can be squared to form a quadratic whole number. For example, 2 X 2 =4; 3 x 3 = 9; -2 x -2 = 4; -3 x -3 = 9. The nmber being squared is called, the square root, of the quadratic (square) number. Notice, however, that in this domain, all quadratic whole numbers are positive. A pradoxical question arises: “Can one form a negative quadratic number?” Or, conversely, “what is the square root of a negative number?”

The simplest case, which subsumes all others, is the case of the quadratic unity, 1. 1 x 1 = 1 -1 x -1 = 1. The square root of 1 = 1 or -1, as both numbers squared equal 1. What, then is the square root of -1?

Within the concept of number as one-dimensional, (two directions), the concept of the square root of -1 remains paradoxical, and was given the unfortunate name of imaginary. (Just as the oligarchy attempted to limit human knowledge to one dimension, by naming the Lydian interval the “devil’s” interval.) Euler and others, sought to limit progress of human knowledge, by giving the square root of -1 a purely formal, and therefore meaningless, definition. It was Gauss, who saw with his mind, in this paradox, a means of extending the concept of number, into a new domain–the complex domain. Instead of avoiding the paradox presented, by thinking of it as imaginary, or impossible, Gauss asked, in what higher domain, must such a magnitude exist? A shift in hypothesis, which his student, Riemann, would later designate as from n to n + 1 dimensions.

Gauss elaborates his hypothesis of the complex domain, in a section of the second paper on biquadratic residues in 1832, but, as he says, he developed the hypothesis, as early as 1799, while writing his original work on higher arithmetic, Disquisitiones Arithmeticae. He, says he was only waiting (over 30 years) for a suitable place, in which to announce his new hypothesis to the public.

Gauss approached the paradox of the square root of -1, by extending the hypothesis underlying the concept of number from one dimension (two directions) to two dimensions (four directions).

For purposes of brevity, the square root of -1 is denoted by the letter i. (Again, an unfortunate designation, associated with the term imaginary, owing to the fraudulent Euler.)

Now reflect on the properties of the complex domain, as investigated by Gauss.

In the complex domain of two dimensions, the square root of -1 is thought of as a different dimension, distinct from the dimension of simply positive and negative, but united in the complex domain. In the complex domain, all numbers are made up of two dimensions. One dimension is associated with positive and negative, the other dimension, is associated with +i and -i. The complex domain is ONE domain, indivisible, of two dimensions. A new hypothesis, under which, the positive and negative, i and -i are made congruent (harmonic) with each other. In this new domain, all numbers are of the form a + bi, where a (short for 1 x a) designates the positive-negative dimension, and bi designates the +i -i dimension, (1 + i dimensions). This is not a combination of two different numbers, but one number, with two parts. One dimensional numbers, those limited to positive and negative, such as integers, are the special case of complex numbers where b=0. Complex numbers where neither a nor b is 0 are called mixed complex numbers.

Reflect on the difference between the domain of one dimensional numbers and the complex domain.

In the domain of one dimension, unity is two-fold, 1 and -1. In the complex domain, unity is four-fold, 1 and -1, i and -i. In one dimension, each number is associated with its opposite, for example, 5 and -5, 2 and -2. Its associated number is formed by multiplying by -1. In the complex domain, with its four-fold unity, each number has four associates, found by multiplying that number by -1, i, -i. For example, a + bi is associated with -b + ai, -a – bi, b – ai. (The reader can confirm this for himself, by multiplying a + bi by i, -1, -i respectively.)

There is a special, unique, relationship in the complex domain– the relationship between a number a + bi, and its conjugate, a – bi, that is, when the sign of i is reversed. The product of a number and its conjugate is a^2 + b^2 and is called its Norm. (Notice the similarity to the Pythagorean)

Gauss then investigates the nature of prime numbers in the complex domain. Just as in one dimension, all whole numbers are either prime or composite. However, not all one dimensional prime numbers, remain prime in the complex domain. For example, in the complex domain 2 = (1 + i)(1 – i); 5 = (1 + 2i)(1 – 2i), 13 = (3 + 2i)(3 -2i). In fact, Gauss showed, that all one dimensional prime numbers of the form 4n+1 are no longer prime in the complex domain, but all one- dimensional prime numbers of the form 4n+3 remain prime. (All one-dimensional prime numbers are either of the form 4n+1 or 4n+3. Not all numbers of this form are prime. The reader should verify this himself.)

There are also new kinds of prime numbers in the complex domain–mixed complex prime numbers. Gauss showed that mixed complex numbers are prime, if their Norm is a one-dimensional prime number. For example, 1 + 2i, is a mixed complex prime number, because its Norm, 1^2 + 2^2 = 5, which is a one dimensional prime number. So is 1 – 2i.

Reflect further on the difference between the one dimensional domain of positive and negative numbers, and the complex domain. The complex domain, is not simply the one-dimensional domain, in two directions, as in the fraud perpetrated by Cauchy. IT IS AN ENTIRELY DIFFERENT DOMAIN, lawfully connected, but distinct from the one-dimensional domain. In the complex domain, the universal characteristic is changed. Fundamental singularities, such as prime numbers, are re-ordered. Some are changed, some are unchanged, and new ones are created. It is the domain, which determines the singularities. Is 5 a prime number? Yes and No. It depends on the domain. How do you know what domain you’re in? Through the creative powers of your mind. Analysis situs. If 5 is a prime number, you’re in one dimension, if not, you may be in the complex domain, but, maybe not. Gauss speculates about the possibility of numbers of higher dimensions than two. What happens to prime numbers in domains of more than two dimensions?

Isn’t this the key to improving performance in sales and intelligence?

Mind Over Mathematics–Prime Numbers, Part III

CAN YOU SOLVE THIS PARADOX

Over the previous two weeks, we’ve demonstrated, by reliving a discovery made by the young Carl Friedrich Gauss when he was 10 years old, how numbers are creations of the mind. Once this basic principle is understood, the mind is no longer a slave to formal rules concerning numbers. Problems such as adding all the numbers from 1 to 100, which at first appear tedious and perhaps even difficult, are easily solved, once the mind breaks the formal rules, and re-orders the numbers according to a new, higher principle.

This week, we’ll look more deeply into the nature of numbers, and in doing so, we’ll gain an increased mastery over our minds’ creative process.

What Are Prime Numbers?

Among the whole numbers, there exist unique integers known as Prime Numbers, which are distinguished by the property that they are indivisible by any other number except themselves and 1. Thus, 2, 3, 5, 7, and 11 are all examples of prime numbers. Composite Numbers are integers which can be divided, not only by themselves and 1, but by some other number.

It had already been discovered by the ancient Greeks, and written down by Euclid (flourished c. 300 B.C.) in his “Elements,” that all numbers are either prime or composite, and that any composite number is divisible by some prime number.

You can prove this for yourself, in the following way. Any composite number can by definition be divided by some other number, and that other number is either another composite number or a prime number. If it is a prime number, we need go no further. If it is a composite number, then that new composite number can be divided by another number, which is either a prime number or a composite number, and so on. By this method, you will eventually get to a prime number divisor.

For example, 30 is a composite number, and can be divided into 2, a prime number, and 15, a composite number. In turn, 15, can be divided into 3, a prime number, and 5, also a prime number. So the composite number 30 is made up of, and can be divided by, prime numbers 2, 3, and 5.

A method for discovering which integers are prime, was developed by Eratosthenes in approximately 200 B.C. This approach, known as the Sieve of Eratosthenes. is a method for eliminating the composite numbers from a group of integers, leaving only the primes.

List the integers from 1 to any other arbitrary integer A. Now, beginning with 2 (the first prime number after 1), strike from the list all numbers divisible by 2, for they are composite numbers. Do the same with all numbers divisible by 3, the next prime number; then those divisible by 5, etc., until you come to the first prime number whose square is greater than A. (If A = 100, then you only need do this procedure with primes less than 11.) See Figure 1.

This method allows us to find the prime numbers which are smaller than any arbitrary number–but is there some large number after which there are no prime numbers? In other words, are there an infinite number of prime numbers?

That the answer is yes, is easily proved by showing that, no matter how many prime numbers are found, there can always be found one more. First, find all the prime numbers less than any arbitrary number, a, b, c, …, z. Now multiply all these prime numbers together, and add 1 to the product [(a x b x c x d x … x z) + 1]. Call this new number A. If this new number A is a prime number, then you’ve found another prime, which was not known before. If it is a composite number, then it must be divisible by some prime number. But, that prime number cannot be one of those already known, as the known prime numbers (a, b, c, …, z) will always leave a remainder of 1, when divided into A.

Primes Are Nonlinear

After we have found a large number of primes, it is obvious that, in the small, there is no regular, or linear, pattern of distribution of the prime numbers. Gauss showed, however, that over a large interval, the distribution of the primes is approximated by a logarithmic curve; that is, the larger the prime numbers become, the more spread apart they tend to be. This approximation breaks down increasingly, the smaller the interval. A nightmare for Leonhard Euler–nonlinearity in the small. (See Figure 2.)

{Again, what are Primes?} It is easy to see that any composite number can be decomposed into prime numbers, by division. For example, 12 can be decomposed into 2 x 2 x 3, or 2^2 x 3. The number 504 can be decomposed into 2 x 2 x 2 x 3 x 3 x 7, or 2^3 x 3^2 x 7.

Gauss was the first to prove Disquisitiones Arithmeticae, Article 16) that a composite number can be decomposed into only one combination of prime numbers. In the above examples, no combination of prime numbers other than 2 x 2 x 3 will equal 12. Likewise for 504, or any other composite number.

This remarkable result, which Gauss says was “tacitly supposed but had never been proved,” provokes a fundamental question concerning the nature of the universe. The fact that Gauss was the first to consider this result important enough to prove, is another indication of his genius.

With Gauss’s proof, and the preceding discussion, it is shown that prime numbers are that from which all other numbers are composed. The primes are primary. The word the ancient Greeks used for “prime,” was the same word they used for “first” or “foremost.”

This raises the question, what happens when you try to construct all integers from the primes alone? First, you’d make all the integers composed only of 2, such as 4, 8, 16, …. Then you’d make all the integers composed only of 3, and of combinations of 2 and 3, such as 6, 9, 12, …, and so forth with 5, etc. As you can see, this process would eventually generate all the integers, but in a nonlinear way.

Compare that process with constructing the integers by addition. Addition generates all the integers sequentially, by adding 1, but does not distinguish between prime numbers and composite numbers.

The unit 1 is indivisible, with respect to addition. With respect to division, the prime numbers are indivisible. Both processes will compose all the integers, but that result coincides only in the infinite. In the finite, they never coincide. The difference is between the mental act of addition, and the mental act of division. Don’t try to resolve the matter, by asking if division is superior to addition. Instead, reflect on that which is different between the two processes, the “in-betweenness.” It is the relations between the numbers, which is the object of our thought, not the numbers in themselves.

This anomaly is a reflection of the truth that there exists a higher hypothesis which underlies the foundations of integers–a hypothesis which is undiscoverable if limited to the domain of simple linear addition. By reflecting on this anomaly, we begin, as Socrates says, “to see the nature of number in our minds only” (from Plato’s “Republic”). Our minds ascend, as Socrates indicates, to contemplate the nature of true Being. We ask, “If the domain of primes is that from which the integers are made, what is the nature of the domain from which the primes are made?”

View the above result from the standpoint of Leibniz’s “Monadology”:

“29. Knowledge of necessary and eternal truths, however, distinguishes us from mere animals and grants us {reason} and the sciences, elevating us to the knowledge of ourselves and of God. This possession is what is called our reasonable soul or {spirit.}

“30. By this knowledge of necessary truths and by the abstractions made possible through them, we also are raised to {acts of reflection} which enable us to think of the so-called {self} and to consider this or that to be in us. Thinking thus about ourselves, we think of Being, Substance, the Simple and the Composite, the Immaterial, and even of God, conceiving what is limited in us as without limit in Him. These acts of reflection furnish the principal objects of our reasoning.”

Next week: When is 5 not a prime number.

Prime Numbers, Part IV

CAN YOU SOLVE THIS PARADOX?

“The natural, sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality. So much, indeed, is number shown to be the beginning of those things which are attained by rationality, that with its sublation, nothing remains at all, as is proven by rationality. And if rationality unfolds number and employs it in constituting conjectures, that is not other than if rationality employs itself and forms everything in its highest natural similitude, just as God, as infinite mind, in His coeternal Word imparts being to things. There cannot be anything prior to number, for everything other affirms that it necessarily existed from it….

“The essence of number is therefore the prime exemplar of the mind. For indeed, one finds impressed in it from the first trinity or the unitrinity, contracted in plurality. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

–Nicolaus of Cusa “On Conjectures”

Last week, we investigated the nature of prime numbers. This week, we further unfold the essence of number, requiring our minds to lift themselves into a new, higher domain. This journey may be difficult, at points, for the reader. For those with previous mathematical training, infected by the Euler-Lagrange-Cauchy fraud, you will find your previous training an annoying distraction. For those without such annoying distractions, you may find the lack of such to be an annoying distraction, in itself. To avoid these distractions, follow the prescription of Plato, Cusa, and Leibniz, and see with your mind only. For this purpose, we continue our investigation using the principles of higher arithmetic as developed by Carl Friedrich Gauss, which considers only the relations among whole numbers.

In the previous discussion, we discovered, through reflection on the Prime Numbers, a hypothesis underlying the essence of number. By its extension, we will be led to a new, higher hypothesis underlying the essence of number.

First, extend the idea of number from positive whole numbers, to include their opposites, the negative whole numbers. Here, prime numbers maintain their same relationship with respect to all numbers, with the exception of a change of direction from positive to negative. Whereas positive whole numbers are formed sequentially by adding one, negative whole numbers are formed sequentially by subtracting 1, or adding -1. Positive composite numbers are formed by multiplying the prime factors, negative composite numbers are formed by multiplying the prime factors by -1. Think of positive and negative, not as position, but as directions, opposite to one another. If positive is right, negative is left. If positive is up, negative is down. One dimension–two directions.

In short, we have extended our concept of number, by conceiving of a twofold unity: 1 and -1.

An Anomaly Appears

Once our concept of number is extended into the negative direction, an anomaly immediately appears. All positive and negative whole numbers can be squared to form a quadratic whole number. For example, 2×2=4; 3×3=9; -2x-2=4; -3x-3=9. The number being squared is called the square root of the quadratic (square) number. Notice, however, that in this domain, all quadratic whole numbers are positive. A paradoxical question arises: “Can one form a negative quadratic number?” Or, conversely, “What is the square root of a negative number?”

The simplest case, which subsumes all others, is the case of the quadratic unity, 1: 1×1=1, and -1x-1=1. The square root of 1=1 or -1, as both numbers squared equal 1. What, then, is the square root of -1?

Within the concept of number as one-dimensional (two directions), the concept of the square root of -1 remains paradoxical, and was given the unfortunate name of imaginary. (Just as the oligarchy attempted to limit human knowledge to one dimension, by naming the Lydian interval the “devil’s interval.”) Euler and others sought to limit the progress of human knowledge, by giving the square root of -1 a purely formal, and therefore meaningless, definition. It was Gauss who saw with his mind, in this paradox, a means of extending the concept of number, into a new domain–the complex domain. Instead of avoiding the paradox presented, by thinking of the square root of -1 as imaginary, or impossible, Gauss asked, in what higher domain must such a number exist? In other words, a shift in hypothesis, which his student, Bernhard Riemann, would later designate as changing from {n} to {n}+1 dimensions.

Gauss elaborated his hypothesis of the complex domain, in a section of the second paper on biquadratic residues in 1832, but, as he says, he developed the hypothesis as early as 1799, while writing his original work on higher arithmetic, Disquisitiones Arithmeticae. He says he was only waiting (over 30 years) for a suitable place in which to announce his new hypothesis to the public.

Extending The Hypothesis

Gauss approached the paradox of the square root of -1, by extending the hypothesis underlying the concept of number from one dimension (two directions) to two dimensions (four directions).

For purposes of brevity, the square root of -1 is denoted by the letter {i.} (Again, an unfortunate designation, associated with the term imaginary, owing to the fraudulent Euler.)

Now reflect on the properties of the complex domain, as investigated by Gauss.

Gauss conceived of the complex domain as a domain of two dimensions, in which the square root of -1 is thought of as a different dimension, distinct from the dimension of simply positive and negative, but united in the complex domain. If the domain of positive and negative numbers is thought of as a one-dimensional series, the complex domain can be thought of as a series of one-dimensional series. (See Figure 1.) Movement within a series is associated with the concept of positive and negative. Movement from one series to the next, is associated with +{i} and -{i}.

In the complex domain, all numbers are made up of two dimensions. One dimension is associated with positive and negative; the other dimension is associated with +{i} and -{i}. The complex domain is {one} domain, indivisible, of two dimensions. A new hypothesis, under which the positive and negative, {i} and -{i}, are made congruent (harmonic) with each other. In this new domain, all numbers are of the form {a}+{bi}, where {a} (short for 1x{a}) designates the positive-negative dimension, and {bi} designates the +{i}-{i} dimension, (1+{i} dimensions). This is not a combination of two different numbers, but one number, with two parts. One-dimensional numbers, those limited to positive and negative, such as integers, are the special case of complex numbers where {b}=0. Complex numbers where neither {a} nor {b} is 0, are called mixed complex numbers.

One Dimension, and Complex Domain

Reflect on the difference between the domain of one-dimensional numbers and the complex domain.

In the domain of one dimension, unity is twofold, 1 and -1. In the complex domain, unity is fourfold, 1, -1, {i,} and -{i}. In one dimension, each number is associated with its opposite, for example, 5 and -5, 2 and -2. Its associated number is formed by multiplying by -1. In the complex domain, with its fourfold unity, each number has four associates, found by multiplying that number by -1, {i,} -{i}. For example, {a}+{bi} is associated with -{b}+{ai}, -{a}-{bi}, {b}-{ai}. (The reader can confirm this for himself, by multiplying {a}+{bi} by {i}, -1, -{i}, respectively.)

There is a special, unique relationship in the complex domain–the relationship between a number {a}+{bi}, and its conjugate, {a}-{bi}, that is, when the sign of {i} is reversed. The product of a number and its conjugate is {a}2+{b}2, and is called its Norm.

Gauss then investigated the nature of prime numbers in the complex domain. Just as is the case in one dimension, all whole numbers are either prime or composite, in the complex domain. However, not all one-dimensional prime numbers remain prime in the complex domain. For example, in the complex domain 2=(1+{i})(1-{i}); 5=(1+2{i})(1-2{i}), 13=(3+2{i})(3-2{i}). In fact, Gauss showed that all one-dimensional prime numbers of the form 4{n}+1 are no longer prime in the complex domain, but all one-dimensional prime numbers of the form 4{n}+3 remain prime. (All one-dimensional prime numbers are either of the form 4{n}+1 or 4{n}+3. But, not all numbers of this form are prime. The reader should verify this himself.)

There are also new kinds of prime numbers in the complex domain–mixed complex prime numbers. Gauss showed that mixed complex numbers are prime, if their Norm is a one-dimensional prime number. For example, 1+2{i} is a mixed complex prime number because its Norm, 12+22=5, is a one-dimensional prime number. So is 1-2{i}.

Reflect further on the difference between the one-dimensional domain of positive and negative numbers, and the complex domain. The complex domain is not simply the one-dimensional domain in two directions, as in the fraud perpetrated by Cauchy. {It is an entirely different domain,} lawfully connected, but distinct from the one-dimensional domain. In the complex domain, the universal characteristic is changed. Fundamental singularities, such as prime numbers, are re-ordered. Some are changed, some are unchanged, and new ones are created. It is the domain which determines the singularities.

Is 5 a prime number? Yes and No. It depends on the domain. How do you know what domain you’re in? Through the creative powers of your mind. If 5 is a prime number, you’re in one dimension; if not, you may be in the complex domain. But, maybe not, as Gauss speculates about the possibility of numbers of dimensions higher than two.

For further reading see, in English translation, Gauss’s “The Metaphysics of Complex Numbers,” 21st Century Science and Technology magazine, spring 1990.

## Change is All Ye Know On Earth–And in Heaven

by Bruce Director

It is one thing to re-utter Heraclites’ fragment, “On those who enter the same river, ever different waters flow”, as the aphorism,“Nothing is constant but change,” as it has now become known, but it is entirely different, and significantly more important, to know the meaning of what Heraclites spoke. Even those who regard themselves as cognoscenti fail to understand this, as more often than not, they mistakenly consider change to be ephemeral. Competence in science, and more generally, sanity, depends upon being free from this error, and the mental obsessions that arise from it. However, such liberation comes not through formal abstract contemplations, which are intrinsically vulnerable to Eleatic-type sophistries, but, as Heraclites’ original formulation implies, through a pedagogical confrontation with the real, physically changing universe itself–a confrontation that all humans are, fortunately, blessed to experience. Continue reading Riemann for Anti-Dummies: Part 69 : Change is All Ye Know On Earth — And in Heaven

August 28, 2006

## An Insider’s Guide to the Universe

by Bruce Director

Though all humans are blessed to spend eternity inside the universe, many squander the mortal portion, deluded they are somewhere else. These assumed “outsiders” acquire an obsessive belief in a fantasy world whose nature is determined by {a priori} axiomatic assumptions of the deluded’s choosing, and an insistence that any experimental evidence contradicting these axioms must be either disregarded, or, if grudgingly acknowledged, determined to be from “outside” their world. Typical of such beliefs are the notions of Euclidean geometry, empiricism, positivism, existentialism, or that most pernicious of pathologies afflicting our culture today: Baby-Boomerism.

## Riemann for Anti-Dummies: Part 67 : A View From The Top

August 28, 2006 (6:25pm)

## The View From the Top

by Bruce Director

For more than three millennia the motion of a spinning top has been a source of great amusement for children, scientists, and philosophers. A careful examination of its motion provides insight into the underlying dynamics of the universe and exposes the fraud of absolute space. Plato took great delight in the embarrassment the top’s motion caused his Eleatic and Sophist adversaries, who argued that motion and change did not really exist. Nicholas of Cusa enjoyed the image of a spinning top as a beautiful expression of the universe’s self-boundedness. From a Riemannian standpoint, that same simple spinning top is still a great source of fun, unfolding Cusa’s concept into the domain of hypergeometries, and twisting Plato’s modern adversaries into gnarled knots. Continue reading Riemann for Anti-Dummies: Part 67 : A View From The Top

## On the Subject of Metaphysics

###### By Lyndon H. LaRouche, Jr.

March 18, 2006

As I was reminded by a discussion-partner during the course of this past week, I am among the relatively few, surviving exceptions to an epidemic loss of actual knowledge of even the rudiments of metaphysics among presently living generations. Some hours after that conversation of the past week, Bruce Director’s long-promised draft introduction to the subject of the Arithmetic-Geometric Mean reached me on my current brief tour in Europe. This latest pedagogical by Bruce comes near to the core of a competent illustration of the practical meaning of metaphysics as such. I thought it useful to add the following bit of prefatory spice to Bruce’s work which follows immediately here. Continue reading Riemann for Anti-Dummies: Part 66 : Gauss’s Arithmetic-Geometric Mean: A Matter of Precise Ambiguity

## Riemann for Anti-Dummies: Part 65 : On the 375th Anniversary of Kepler

November 16, 2005

## On the 375th Anniversary of Kepler’s Passing

by Bruce Director

“In anxious and uncertain times like ours, when it is difficult to find pleasure in humanity and the course of human affairs, it is particularly consoling to think of the serene greatness of a Kepler. Kepler lived in an age in which the reign of law in nature was by no means an accepted certainty. How great must his faith in a uniform law have been, to have given him the strength to devote ten years of hard and patient work to the empirical investigation of the movement of the planets and the mathematical laws of that movement, entirely on his own, supported by no one and understood by very few! If we would honor his memory worthily, we must get as clear a picture as we can of his problem and the stages of its solution.” Continue reading Riemann for Anti-Dummies: Part 65 : On the 375th Anniversary of Kepler

## Hypergeometric Harmonics

by Bruce Director

In the same year that Riemann published his {Theory of Abelian Functions}, he also produced a companion piece of equal significance titled {Contributions to the Theory of Functions representable by Gauss’s Series F(a,b,c,x). The content of these works was the polished product of material Riemann developed in a series of lectures delivered at Goettingen University during the 1855-56 interval and through earlier discussions with Gauss and Dirichlet. Continue reading Riemann for Anti-Dummies: Part 64 : Hypergeometric Harmonics

## Riemann For Anti-Dummies Part 63: Dynamics not Mechanics

by Bruce Director

Despite the prevailing popular opinion to the contrary, human beings are not mechanical systems. So, if you wish to begin to understand the science of physical economy, you must know the science of dynamics, as distinct from, and superior to, the Aristotelean sophistry called mechanics.

The distinction between dynamics and mechanics is not semantic. It is fundamental. Dynamics concerns causes. Mechanics concerns effects. Dynamics treats processes as a whole. Mechanics treats the interaction among some of its parts. They are as different as truth from rhetoric, ideas from words or, love from sex. Though this distinction arises directly through an investigation in the domain of physical science, its implications, as Leibniz himself emphasized, are universal. For without an understanding of the dynamics of situation, it is impossible to know anything about politics, history, science or art. Continue reading Riemann for Anti-Dummies: Part 63 : Dynamics not Mechanics

## On the Continuum of the Discontinuum

by Bruce Director

In the Laws, Plato’s Athenian stranger laments that the Greek people’s pervasive ignorance of the incommensurability of a line with a surface and a surface with a solid, had rendered them more like “guzzling swine” than true human beings. While the same lament can be sounded with respect to today’s society, modern citizens must also include an understanding of those higher transcendentals which are the subject of Riemann’s “Theory of Abelian Functions”, if they are to avoid the lamentable fate of Plato’s fellow Greeks. To rectify this condition, Plato recommended that these subjects should be learned in childhood, through play. Continue reading Riemann for Anti-Dummies: Part 62 : On the Continuum of the Discontinuum

## Riemann for Anti-Dummies: Part 61 : To What End Do We Study Riemann’s Investigation of Abelian Functions?

March 5, 2005 (8:08pm)

## To What End Do We Study Riemann’s Investigation of Abelian Functions?

by Bruce Director

On June 10, 1854, Bernhard Riemann shocked the world by stating the obvious:
For more than two thousand years scientists had accepted, as dogma, that the axioms of Euclidean geometry were the only foundation for science, despite the fact that science had been left in the dark as to whether these axioms had any physical reality. “From Euclid to Legendre, to name the most renowned writers on geometry, this darkness has been lifted neither by the mathematicians nor by the philosophers who have labored upon it.” The axioms of Euclidean geometry are, Riemann said, “like all facts, not necessary but of a merely empirical certainty; they are hypotheses; one may therefore inquire into their probability, which is truly very great within the bounds of observation, and thereafter decide concerning the admissibility of protracting them outside the limits of observation, not only toward the immeasurably large, but also toward the immeasurably small.” Continue reading Riemann for Anti-Dummies: Part 61 : To What End Do We Study Riemann’s Investigation of Abelian Functions?

## The Power to Change, Change

New ideas, like people, come into this world naked. To effectively perform their mission, they must be provided with clothes. But unlike children who can speak for themselves, ideas must be dressed in words and images which, upon careful reflection, indicate how they were conceived. And though it would be a grave error to mistake a person’s substance for his or her outward appearance, the spirit of an idea (which also cannot be captured by a superficial account of its form) can, nevertheless, be evoked by that form’s animation. Yet there are those for whom the generation of such thoughts had seemed impossible, and to whom the very existence of such creations signifies a power they had denied could be. They focus only on the form, gossiping about its appearance, chiding its unconventionality or smothering it with an effusive description of its external features.

The history of science is replete with examples of such new creatures, which have been defended from such sophistries by their authors’ careful constructions: Archytas’s solution for the doubling of the cube, or Gauss’s complex surfaces, to name but two of those reviewed in previous installments of this series. Though such constructions arise as the unexpected solutions to specific problems, they embody the powerful new thoughts which made such solutions possible. Thus, their recreation evokes that more general principle, which, though forever connected to its origins, emerges as a new universal idea, and gains for all time, a universal increase in the cognitive power of man.

Now we turn our attention to another such creation, previously mentioned, but not yet adequately developed in this forum: the hypothesis which lies at the foundations of Riemann’s surface.

Riemann first presented to the world his new idea in his doctoral dissertation of 1851, and elaborated its implications in his 1854 habilitation lecture, his 1857 treatises on Abelian and hypergeometric functions, and his posthumously published philosophical fragments. From all these sources, and the historical context in which they were produced, we can reconstruct Riemann’s new idea as the solution to the unresolved physical paradoxes brought to the fore by C.F. Gauss’s extension of Leibniz’s calculus into the complex domain.

But from this exercise we acquire much more. As we form Riemann’s surfaces in our mind as images, we begin to recognize the quality of mind which produced this solution, and a more universal thought takes shape as well. From this point forward these images evoke in our minds that universal thought. Thus, we can bring Riemann’s creation to life, not simply as a solution to a formal mathematical problem, as its outward appearance is most frequently portrayed, but as an expression of an epistemological concept that has revolutionized human thinking.

What is a Surface?

When Riemann and Gauss speak of surfaces, they do not mean visible objects embedded in a linearly-extended Euclidean-type space. Rather, they speak of what Riemann identified in his 1854 habilitation paper as “multiply-extended continuous manifolds.” Such manifolds are not defined by a set of a priori axiomatic assumptions, but are concepts arising from an investigation of physical action determined by universal physical principles.

“In a concept whose various modes of determination form a continuous manifold, if one passes in a definite way from one mode of determination to another, the modes of determination which are traversed constitute a simply extended manifold and its essential mark is this, that in it, a continuous progress is possible from any point only in two directions, forward and backward. If now one forms the thought of this manifold again passing over into another entirely different, here again in a definite way, that is, in such a way that every point goes over into a definite point of the other, then will all the modes of determination thus obtained form a doubly extended manifold. In similar procedure one obtains a triply extended manifold when one represents to oneself that a double extension passes over in a definite way into one entirely different, and it is easy to see how one can prolong this construction indefinitely. If one considers his object of thought as variable instead of regarding the concept as determinable, then this construction can be characterized as a synthesis of a variability of “n + 1” dimensions out of a variability of “n” dimensions and a variability of one dimension.” (Riemann’s Habilitation Lecture)

Before continuing with this more general investigation of multiply extended manifolds, let us take as an example, the specific cases of the catenary and the catenoid.

As was illustrated in the last installment of this series, Leibniz, through his infinitesimal differential calculus, created a means to express physical action as the intended effect of a universal physical principle that is acting, universally, but differently, at every infinitesimal interval of time and space. In the case of the catenary, this is expressed by the fact that the visible shape of the catenary is determined by the changing effect of the physical principle of least-action on a chain supporting its own weight. Though this effect is different at each point along the catenary curve, these differences are determined by that general principle which integrates them so as to produce a physically stable chain.

Now, compare this case with that of the surface of a catenoid, formed, for example, by a soap film suspended between two circular hoops. (See Figure 1.)

 Figure 1

Here the soap film is suspended between the hoops along catenary curves, but this family of catenaries are themselves integrated, along circular pathways, into a surface. Thus, like the catenary, the visible shape of the catenoid is determined by the changing effect of the physical principle of least action, but instead of those changes varying only along one curve, as in the case of the catenary, those differences occur within a rotational manifold of catenary curves. Therefore, the general principle that determines the stable shape of the soap film is expressed as the integrated effect that unites these two distinct but connected types of differences under one, higher principle.

From the standpoint of Riemann’s habilitation lecture cited above, the catenary comes under his concept of a simply extended manifold, whereas, the catenoid is a type of doubly extended manifold. Using Leibniz’s calculus, the changing effect of least-action can be expressed geometrically, by a type of animation called a differential equation. In the case of a simply extended manifold, the changing effect of least action is expressed as the changing curvature of a curve, and for a doubly extended manifold, the changing curvature of a surface. Thus, in both cases, the visible form of the action is expressed as a function of the changing effect of the universal principle of least action.

It must be underscored that in both cases, the visible shape of the curve or the surface is the effect of the characteristic {physical} “modes of determination” of the manifold. These modes of determination are not visible, but they are faithfully reflected in the visible effects. The challenge for science is to be able to measure these effects, and from their variations in the infinitesimally small, determine the principles that integrate them into a unified extended manifold of action.

As Riemann stated in his lecture notes On Partial Differential Equations and their Applications to Physical Problems:

“What has been shown to be a fact by induction, that differential equations form the actual foundation of mathematical physics, can also be shown a priori. True elementary laws can operate only in the infinitely small, and apply to points in space and time. But such laws are in general partial differential equations, and the derivation of laws for extended bodies and time periods requires their integration. Thus we need methods for deriving from laws in the infinitely small such laws in the finite, and, more precisely, derived with complete rigor, without permitting oneself omissions. Because it is only then that we can test them by way of experience.”

Riemann was basing his work in this regard on the previous achievements of Gauss, specifically, Gauss’s treatises on curved surfaces, conformal mapping and potential theory, some of which has been discussed in previous installments of this series and some of which will be discussed in future pedagogicals. For purposes of illustrating the concepts essential for this discussion, we return to the case of the catenoid.

Like the catenary, the catenoid is able to maintain its stable shape, because the tension exerted at every point is equalized by the changing effect of the principle of least action. Or, inversely, the shape of the catenary is the unique form which equalizes the tension exerted along the chain, by the effort of the chain to support its weight under the effect of gravity. But, unlike in the catenary where the tension is exerted back and forth along a curve, in the catenoid, the tension is exerted in an infinite number of directions radiating out from every point.

This changing effect that produces equal tension can be expressed, geometrically, by the changing curvature of the catenary curve, that same effect can be expressed, for the catenoid, by the changing curvature of the surface. But here the problem becomes more difficult, because the curvature of the curves within the surface is always different, depending on the curve’s direction.

Gauss solved this problem by defining several ways to measure the curvature of a surface. He recognized that at every point on any surface, though there be an infinite number of curves radiating along the surface from that point, one of them was the most curved and one of them was the least, and that these curves were always perpendicular to each other. (See Figure 2.)

 Figure 2

Gauss measured the curvature of these curves, after Leibniz, by the size of the osculating circles to the curves at that point. (See Figure 3.)

 Figure 3

Since the larger the osculating circle, the smaller the curvature, and vice versa, Gauss defined the measure of curvature at each point, as the inverse of the product of the radii of the extreme osculating circles. The characteristic curvature of a surface could thus be expressed by how this measure of curvature changed over the extent of the surface.

Riemann’s Surfaces

Whereas the catenoid is exemplary of a type of surface determined by a physical principle, such surfaces, in most cases, do not necessarily manifest a visible form. Nevertheless, a non- visible, but physical form can be clearly defined, as Riemann indicates, by generalizing the idea of a surface, to a concept of a multiply-extended manifold of physical action. From this concept of a physical manifold, a visible, geometrical form can be constructed, that faithfully reflects the characteristics of the physical manifold. For example, the surface of the Earth, as Gauss understood it, is the physical manifold that is everywhere perpendicular to the pull of gravity. This doubly extended manifold is measured by the changing effect that gravity has on the direction of a plumb bob. This surface, while not directly visible, is nevertheless physically determined, and thus, defines a physical geometric thought object, whose characteristics reflect the principles of physical action of gravity. Another example investigated by Gauss, is the physical manifold of the Earth’s magnetic effect, which, as a triply-extended manifold, is measured by the changing directions of a compass needle at different positions on the Earth. Here, too, the surface is not visible, but is, nevertheless, a physically determined surface.

Gauss called these types of non-visible physical surfaces “potential” surfaces, because they express the potential for action of a physical principle. These manifolds do not express what is {visible}. Rather they express what is {possible}, with respect to the physical constraints imposed by a universal physical principle on the manifold of action, just as the physical principle of least action imposes a certain characteristic curvature on a hanging chain or soap film. Thus, when investigating any physical process, it is necessary to form a concept of the manifold of physical principles in which that process is occurring. These principles impose a characteristic potential for action, which, Leibniz, Gauss and Riemann showed, can be expressed, geometrically as a characteristic curvature of the manifold. (Riemann’s Dirichlet Principle Riemann for Anti- Dummies Part 58.)

Following Gauss, Riemann recognized that in the type of least action physical manifold exemplified by the catenoid or Gauss’s potential surfaces, the curves of maximum and minimum curvature are harmonically related, which means that their mutual curvatures change at the same rate, in perpendicular directions. For example, in the catenoid, the size of the osculating circles of maximum curvature and minimum curvature increase or decrease at the same rate when moving from place to place on the surface.

It should again be emphasized, that this harmonic relationship is an effect, not a cause. The principle of least action is the cause, which, in the visible domain is reflected in this harmonic relationship.

Gauss had recognized in his Copenhagen Prize Essay, that this harmonic relationship, between the curves of maximum and minimum curvatures of a surface, is a characteristic of functions in the complex domain. Gauss applied this discovery by showing that the problem of mapping one surface onto another, so that all angular relationships are preserved, could be solved by finding a complex function that transformed one set of harmonically related curves into another.

Riemann extended Gauss’s idea by showing that from such complex functions, an entire class of physical manifolds can be expressed, whose “harmony would otherwise have remained hidden.” For Riemann, complex functions were a means to express a transformation from one physical manifold into another, as an effect of changing the curvature of the manifold’s determining characteristics. Further, as we will soon see from the implications of the essential features of Riemann’s surfaces, Riemann’s generalization of Gauss enabled him to express a new type of physical transformation, in which the potential for action is not simply transformed, but is increased in power.

Or, in other words, Riemann’s complex functions could express the power to change, change.

To pedagogically illustrate Riemann’s idea we will first introduce it, clothed in the words and images of geometry. From there, its more universal implications can more easily be brought to light.

This is a method similar to that used in investigating Archytas’s solution for doubling the cube or Gauss’s solutions to the fundamental theorem of algebra. In both cases the requirements of a specific geometric solution reveal that the original paradox was ontological, not mathematical, as it had first been presented. For example, the fact that in order to double the cube, Archytas had to produce the multiply connected manifold of physical action that generates, in a single act, the torus, cylinder and cone, demonstrates that the principle that generates the cube is outside the apparently spherical boundary of visible space.

Similarly, as Gauss showed in his proof of the fundamental theorem of algebra, the apparent mathematical paradox of the ?-1, was, in reality, not a problem of algebra, but a problem of algebraists, who were fixated on a false conception of the physical universe. As Gauss noted, the paradox of the ?-1, is not a formal mathematical one, even though it may have appeared in that form. Rather these paradoxes actually reflect “the deepest questions of the metaphysics of space”, as Gauss himself said.

Riemann exploited one such mathematical paradox to force to the surface his new conception concerning manifolds of physical action.

That paradox arises , in its geometrical context, when investigating complex functions represented, as Gauss did in his Copenhagen Prize Essay, as conformal transformations of one surface onto another. In some cases, these mappings are quite straightforward, such as the stereographic projection of a sphere onto a plane, or the conformal mapping of the spheroidal shape of the Earth onto a sphere. In these cases, there is a one to one correspondence between every point of one surface and every point of the other.

However, in some complex functions (most emphatically, all those transcendental functions that correspond to physical action), this one to one relationship does not occur. For example, the complex squaring function, maps the points of one surface onto another twice. Or, in the case of a complex cubic function, the points of one surface are mapped onto another three times.

This can be illustrated with an animation (See Figure 4a.) representing a complex cubing function.

 Figure 4a

Since the complex cubing function triples the angle of rotation, a 120 degree rotation of the surface of the circular disk on the left maps to a full rotation of the surface of the circular disk on the right. Likewise, the second 120 degree rotation of the surface of the circular disk on the left also maps to a full rotation of the circular disk on the right. So to with the third 120 degree rotation. Thus, each point of the disk on the right, corresponds to three points on the disk on the left. This inversion can be seen more clearly in Figure 4b which maps the disk on the left as a function of the action of the disk on the right.

 Figure 4b

Thus, there appears to be a mathematical ambiguity because one clearly defined action produces three distinct effects, and there is no formal algebraic way to distinguish one of these effects from the other. However, as Gauss did in the fundamental theorem of algebra, Riemann, applying Leibniz’s method of analysis situs found a physical geometrical form, which uniquely characterized the quality of physical action, that underlies such multi-valued complex functions.

He rejected the method of Cauchy, who sought to define an algebraic calculation to distinguish one effect from the other. Instead, Riemann focused on the unique points of the mapping that were not multi-valued the singularities. As Gauss did in his solution to the fundamental theorem of algebra, Riemann created a construction whose geometric characteristics expressed the organizing principle of the manifold by the relationship between what the singularity and everything else in the manifold.

To visualize Riemann’s idea, we can use the above example of a cubic function. For such a function, Riemann conceived of each of the three different effects produced by the cubic rotation, to be different branches of one action. He represented each as a different layer on a surface of three sheets. These layers were connected to each other at the point of singularity, where their values all coincided. He called this point a branch point. (See Figure 5.)

 Figure 5

Thus, in our example, the effect of the first 120 degree rotations are spread over the first layer, the effect of the second 120 degree rotation is spread over the second layer, and the effect of the third 120 degree rotation is spread over the third layer. But since the action is continuous so must be the effects. To express this, Riemann connected these layers to each other by making a cut emanating outward from the branch point, which he called a branch cut. Along this cut, he connected the first layer to the second, the second to the third and the third back to the first. (See Figure 6.)

 Figure 6

In this way, Riemann expresses a physical relationship in which a single continuous action that produces a multiplicity of effects are integrated into one continuous manifold.

It is important to underscore, that though we can create a vivid image in our mind of Riemann’s surface, it is impossible to physically construct it in visible space. Nevertheless, Riemann’s imagined surface, because it reflects an ontological principle of action, produces a very real effect. From it we can gain a greater insight into the relationship between a manifold of universal physical principles and their physical effects.

The ontological implications of Riemann’s surfaces begin to shine through if we investigate a slightly more complicated example. The next set of animations illustrate Riemann’s surface for a fourth power algebraic function, which produces a Riemann’s surface with four layers. The sets of perpendicular lines on the left and quartic curves on the right represent the effect of the transformation on the manifold’s harmonically related curves of maximum and minimum curvature.

In the first animation, these four roots are represented by four colored dots, that are shifted, in Riemann’s surface to lie on top of each other at the branch point. (See Figure 7.)

 Figure 7

Now, think of this Riemann’s surface as a type of manifold of physical action, and investigate the effect of this transformation on different actions. In Figure 8a we see the effect within this manifold on a circular rotation around one of the roots.

 Figure 8a

The original action is depicted in the left panel and its effect is depicted on the right. In Figure 8b, we see the same action depicted from a side view of Riemann’s surface, showing that the pathway loops around the one layer containing the root and then continuing, without looping through the rest of the layers.

 Figure 8b

In Figure 8b we see top down view of 9b. Compare this with the result in 8a.

In Figures 9a, 9b, and 9c we see a similar representation of the same type of action but around a different singularity, and with a similar effect.

 Figure 9a Figure 9b
 Figure 9c

In Figures 10a, 10b, 10c we see the effect on a closed pathway that encompasses no singularities.

 Figure 10a Figure 10b
 Figure 10c

But, in Figures 11a 11b, and 11c we see a simple closed pathway that circles two singularities.

 Figure 11a Figure 11b
 Figure 11c

Here the effect of the transformation is to produce a double loop around the branch point. Thus, within this type of manifold, an action that encompasses a greater number of singularities produces a greater effect. In other words, Riemann’s surface expresses a type of multiply extended continuous manifold in which the power of any action is a function of the number of singularities encompassed by that action.

In Figures 12a, 12b, 12c we see this power is increased once again when the pathway of action encompasses three singularities.

 Figure 12a Figure 12b
 Figure 12c

And, in Figures 13a, 13b, 13c, a further increase in power is represented by a loop around 4 singularities which is transformed into a quadruple loop around the branch point.

 Figure 13a Figure 13b
 Figure 13c

Here we have reached the limit of the power of this particular type of complex function. To increase the function’s power, a new principle must be introduced, which will add a new singularity to the manifold of physical action. With the incorporation of this new singularity, into the manifold, a greater potential for action is achieved.

This defines a still higher type of transformation, one that adds new singularities to the manifold of action. But in these algebraic examples, these singularities are added one by one. Riemann showed, than Abel’s extended class of higher transcendental functions, when expressed on Riemann’s surface, express a type of transformation that increases the rate and the density at which singularities can be added.

This part of the investigation will have to be continued until next time. But, at least we have begun to scratch the surface.

## Think Infinitesimal

by Bruce Director

“It is well known that scientific physics has existed only since the invention of the differential calculus,” stated Bernhard Riemann in his introduction to his late 1854 lecture series posthumously published under the title, “Partial Differential Equations and their Applications to Physical Questions”. For most of his listeners, Riemann’s statement would have been fairly straight forward, for they understood the physical significance of Leibniz’s calculus as it had percolated over the preceding sesquicentury through the work of Kaestner and Gauss. A far different condition exists, however, for most of today’s readers, whose education has been dominated by the empiricism of the Leibniz-hating Euler, Cauchy and Russell. While such victims might find the formal content of Riemann’s statement agreeable, its true intention would be as obscure to them as the Gospel of John and Epistles of Paul are to Karl Rove and his legions of true believers.

The empiricist will not understand Riemann’s statement, for the simple reason that what he associates with the words “differential calculus” is a completely different idea than what Riemann and Leibniz had in mind. To the victim of today’s empiricist-dominated educational system, the infinitesimal calculus concerns only a set of rules for mathematical formalism. But to the scientist, the infinitesimal calculus is a kind of Socratic dialogue, through which man transcends the limitations of sense-perception and discovers those universal principles that govern all physical action.

The empiricist rejects Leibniz’s notion, because he accepts Aristotle’s doctrine that “physics concerns only objects of sense”, whereas Plato, Cusa, Leibniz and Riemann emphasized, physics concerns objects of {thought}. These thought-objects, or “Geistesmassen” as Riemann called them, refer to the universal principles which {cause} the objects of sense to behave the way they are perceived to behave. Not being directly accessible to the senses, such principles appear to come from “outside” the visible world. However, a great mistake is made if one concludes from this, as the sophists do, that these principles come from outside the universe itself. In fact, these principles, being universal, are acting everywhere, at all times, and in every “infinitesimal” interval of action, osculating the objects of sense as if tangent to the visible domain.

It is this relationship between the observed motions of the objects of sense, and the universal principles acting everywhere on them, that Leibniz’s differential calculus is designed to express. Through it, a universal principle, as it is seen and unseen, is enfolded into a single thought, showing us what is known, and indicating to us what is yet to be discovered. A scientist who turns away, under Aristotle’s, Sarpi’s, or Russell’s, influence, from these objects of thought, to objects of mere sense, is acting as if his own mind has ceased to exist, which, in fact, it has.

Just as Riemann correctly asserts, that scientific physics began with the invention of the differential calculus, it can be justly stated that the differential calculus began with Cusa’s excommunication of Aristotle from science. While it is true that some of the methods of Leibniz’s calculus were beginning to develop in the work of Archytas and Archimedes, this development was arrested when Aristotle’s doctrines became hegemonic in European culture, following the murder of Archimedes by the Romans. Cusa reversed this disaster and reoriented European science away from its obsession with objects of sense, and back to the Pythagorean/Socratic focus on the idea.

Cusa insisted that perception is not caused by sensible things, but that things are sensible because the mind has the power to sense. In turn, the mind is able to sense, because it possesses a still higher faculty of rationality; and it is able to rationalize because it possesses a still higher faculty of intellect; and it is able to intellectualize because man is created in the infinitesimal image of God.

From this standpoint, Cusa rejected Aristotle’s sophism that less change equals greater perfection, which made God a tyrannical force who keeps the world perfect by opposing change. Instead, Cusa recognized that the capacity for change in the physical universe, and in the human mind, indicated the perfectability of both, and that it was God’s intention to perfect his Creation through the cognitive powers of Man. Thus, it is the power of the mind to perceive change, not objects, through which Man relates to the physical world and increases his knowledge of, and power in, the universe.

Having freed science to recognize change as primary, Cusa concluded that all physical action must be non-uniform, which Kepler experimentally validated with his discovery that the principle of universal gravitation produces harmonically-related elliptical orbits. As Kepler insisted, the observed changing motion of the planet in its orbit, rather than an apparent deviation from Aristotle’s illusory idea of fixed perfection, was, in reality, the intended effect of the principle of universal gravitation, which is acting, universally, but whose effect differs, in every infinitesimally small interval of action. In this respect, Kepler likened the principle of universal gravitation to an idea, (using the Latin word {species} to describe it), but distinguishing it from a human idea, because it lacked the quality of willfulness unique to human cognition. Man could grasp this idea, Kepler understood, by forming a concept (thought-object), which expressed the physical action as a consequence of a universal intention, analogous to, as Cusa emphasized, the way human action is the consequence of human intention. While Kepler made significant progress in creating geometrical expressions for this relationship, he recognized the need for a new form of metaphor, and demanded that future generations make further progress to this end.

It was Leibniz who defined the required concept, to which Riemann refers as the beginning of scientific physics. Leibniz recognized that what was required was a new form of mathematical expression, one that expressed the relationship between the universal principle and its changing effect on the observed motion. Most importantly, this new expression must work in reverse, because that is the way it is encountered in scientific investigation. That is, though the effect of the principle is perceived through the motion, merely describing what is observed states nothing about the principle. To scientifically know the cause of the motion, it is necessary to express the motion as an effect of the principle.

To grasp this thought, Leibniz utilized a form of investigation that had been previously developed by Cusa: the relationship of maximum and minimum. As Cusa specified, the maximum and minimum coincide in God, but in the created world the maximum and minimum appear as opposites. Thus, to know any physical process, it is necessary to have a concept through which the opposite extremes of that physical process are recognized as the maximum and minimum effects of a single, unified, intention. For example, in every interval of an elliptical orbit, the planet’s motion is different at the two extremes of that interval, no matter how small an interval is taken. There are but two exceptions. One is the entire orbit, the other is the moment of change itself, which comprise, respectively, the maximum and minimum effect of the principle of universal gravitation on the planet. In the minimum, the universal principle’s effect is always different, but it is differing according to a well-defined principle. The mathematical expression of this differentiation, Leibniz called “the differential” which always exists integrated into the whole action. In the latter form, its mathematical expression, was called by Leibniz, “the integral”.

From this relationship, Leibniz invented a type of animation, which he called “differential equations”, in which the maximum effect is expressed as a function of the minimum. As Riemann noted, this put science on a completely new footing, for in experimental investigations it is the minimum expression that is measured, from which the maximum must be determined, as, for example, in the case of Leibniz’s and Bernoulli’s determination of the catenary, Gauss’s determination of the orbit of Ceres, or, Gauss’s or Riemann’s investigations in geodesy, geomagnetism, electromagnetism and shock waves. With Leibniz’s differential calculus, such investigations could be undertaken, for the first time, with the necessary epistemological rigor.

Of course, Leibniz’s differential equations do not express the principle directly. But, they can express the changing effect of the principle at every moment. On this basis, the principle, can be known by inversion, as that idea which produces the effect expressed by the differential equation. To emphasize the point: the differential equation is not the principle, but it expresses the ever changing footprints that the principle leaves on the visible domain. While this description, clothed in either words or geometry, is necessarily ironic, the thought-object to which it refers, is recognized, in the mind, with absolute precision.

It is crucial to emphasize that Leibniz’s calculus is a mathematical expression of a physical idea. As is obvious with respect to physical action, the differential and the integral express the minimum and maximum effect of the same universal physical principle. Yet the empiricists attacked Leibniz’s calculus by abstracting it from physics and presenting it only as a mathematical formalism. They produced through sophistry, an apparent mathematical paradox, by treating both as if they existed separately from the physical principle they expressed. From this formal mathematical standpoint, the sophist argued, the differential does not exist, because in the moment of change, the time elapsed and distance traversed are both, formally, null. From this, the sophistry continues, the integral can’t be expressed, because it is the sum of infinitely many null magnitudes.

Leibniz countered this sophistry by always insisting on the physical nature of his calculus. In a 1702 letter to Varignon, he posed the following paradox to the algebraists:

Construct two similar triangles from the intersection of two lines. (See Figure 1.)

In the construction, the legs of the large triangle are in the same proportion to each other, as the legs of the smaller one. Now, move the oblique line in a motion parallel to its original position. (See animated Figure 2.)

Under this motion, the smaller triangle gets smaller while the larger triangle gets bigger, but the proportionality of their sides remains the same. At some moment, the smaller triangle passes through the point of intersection of the two original lines, emerging, in the next moment, on the other side, to begin growing again.

The algebraists insisted that, at the moment that the small triangle passed through the point, its sides both appeared to be null, and so it is impossible to express their proportion, or more absurdly, that their proportion ceased to exist at that moment. Leibniz countered that the triangle passed through that point as the result of a physical motion, intended to maintain the proportionality of the sides of the triangles. Consequently, it was the motion that produced the constant proportionality, as its intended effect. At the moment the small triangle passed through the point, the motion did not cease and, thus, neither did the proportionality of the triangles. That proportionality reflected a principle of physical action which is known, in the mind, by a thought-object associated with a certain intention. The point is but a moment in the motion. It does not exist without respect to the physical action. Only when the mathematical expression is separated from the physical action, does the algebraic contradiction appear. The appearance of such a contradiction may indicate a problem with the thinking of the empiricist, but the problem lies only there, not in the physical universe itself.

Conversely, to insist that the algebraic contradiction has an ontological significance, is to induce a state of mental disassociation in the mind of the scientist. This is precisely the intention of Euler, Lagrange, and especially Cauchy, who replaced Leibniz’s idea of the infinitesimal with Cauchy’s idea of the limit. Cauchy argued that the limit removed the algebraic contradiction of the infinitesimal. But in doing so, Cauchy was actually inducing insanity, by removing the connection of the mind to the physical universe in which it exists. This, of course, was his intended effect.

That the cognitive capacity of the mind was the real target of the oligarchy’s attack on Leibniz’s calculus, was confessed to a popular audience by Richard Courant and Herbert Robbins in their English language 1941 book, {What is Mathematics}:

“…the very foundations of the calculus were long obscured by an unwillingness to recognize the exclusive right of the limit concept as the source of the new methods. Neither Newton nor Leibniz coudl bring himself to such a clear-cut attitude, simple as it appears to us now that the limit concept has been completely clarified. Their example dominated more than a century of mathematical development during which the subject was shrouded by talk of `infinitely small quantities’, `differentials’,`ultimate ratios’ etc. {the reluctance with which these concepts were finally abandoned was deeply rooted in the philosophical attitude of the time and in the very nature of the human mind}” (emphasis added, poor punctuation in the original bmd.).

The empiricist sees objects in motion and imagines them to move in a space that is as empty as he believes his own mind to be. A scientist envisions a manifold of universal physical principles, embodied as animated objects of thought that enliven the objects before his eyes. To the former, change is an annoying inconvenience that disrupts his ultimately futile attempts to maintain his accepted axiomatic-formal structure. To the latter, change is the happy indicator of the moving effect of universal principles acting, universally, yet differently, at all infinitesimal intervals of time and space.

The Differential Calculus Animated

The most effective pedagogical means to illuminate Leibniz’s concept of the differential calculus, is through a series of animations that illustrate its application from Kepler, to Huygens, to Leibniz, to Gauss, to Riemann. In what follows, we rely on the animations to do most of the talking, with these written words providing only the barest of stage directions:

Animated Figure 3: Kepler’s principle of equal areas. Kepler conceived of the orbital path as the changing effect of the principle of universal gravitation, which varied inversely to the distance between the sun and the planet. Kepler understood the motion in any interval to be the sum of the infinitely many changing radial distances within that interval, which reflected the planet’s motion at every moment. He could not calculate this sum precisely, but he recognized the result corresponded to the area swept out (See animated figure 3a.),

which he measured through his famous method of the three anomalies. (see animated figure 3b.)

Kepler’s method of calculation led to the paradox which prompted Leibniz to develop his concept of the differential and the integral.

Animated Figure 4: Huygens attempted to tackle the problem of non-uniform motion by expressing one non-uniform motion as a function of another, by the method of involute and evolute. In animated figure 4a the yellow curve is formed by the motion of unwrapping the white string from blue curve.

The yellow curve is called the involute. The blue curve is called the evolute. Thus, the changing curvature of the involute is a function of the changing curvature of the evolute. The white string is always perpendicular to the involute and always tangent to the evolute. Because of this, the involute is the envelope of circles whose centers all lie on the evolute. (See animated figure 4b, 4c.)

In other words, these circles are everywhere tangent (osculating) to the involute, and their radii are everywhere tangent to the evolute. Thus, the curvature of the involute expresses the effect of the principles acting tangentially on the evolute and vice versa.

Now, instead of thinking about these osculating circles being formed by the curves, think of the curves being formed by the osculating action of a circle whose size and position are changing according to a principle of motion. (see animated figure 4d.)

In this way, the curves are more truthfully understood, as the intended effect of a principle of change that is acting, everywhere tangent, to their visible expression.

Huygens used this relationship to build his famous pendulum clock, on the principle that the cycloid had both the property that its involute was another cycloid, and that it was the curve of equal-time for a body falling according to gravity. (See animations 4e,4f,4g,4h.)

Animated Figure 5: While Huygens’s method of the involute and evolute expresses non- uniform motion, it relies on a purely mechanical procedure, instead of expressing a principle of change directly. Leibniz solved this problem by expressing this principle of change through differential equations. To measure the differential, Leibniz projected the changing action in the infinitesimally small into the visible domain, in a manner similar to Plato’s cave metaphor. To do this, Leibniz generalized the investigations of Fermat by defining a series of functions that depended on the changing curvature that resulted from the physical action. (See figure 5a.)

In particular, Leibniz investigated the motion of the subtangent, whose length is a function of the direction of the tangent, which in turn is function of the changing curvature. Leibniz considered the triangle formed between the point of tangency , the intersection of the ordinate with the axis, and the intersection of the tangent with the axis, as a projection into the visible domain, of the changing action in the infinitesimally small.

To get an intuitive grasp of this method, take the example of the parabola (See animated figure 5b )

which illustrates the changing motion of the parabola’s subtangent. Fermat had shown that the subtangent of the parabola is always twice the distance of the abscissa to the vertex. From this standpoint, the parabola can be entirely defined as the effect of a principle of change. Instead of thinking of the subtangent as a function of the parabola, think of the effect of the parabola as a function of a subtangent which is always bisected by its vertex, which, in turn, is defined as the point at which the subtangent is at its minimum. This way of thinking is a very elementary, pedagogical description of a “differential equation”.

From this standpoint, Leibniz was able to discover the existence of physical principles that were not expressed by the visible form of the motion, but {were} expressed through the characteristics of differential equations. For example, the visible shape of the exponential curve can be defined as the curve produced by a continuous motion that is arithmetic in one direction, and geometric in a perpendicular direction. (See animated figure 5c.)

Yet, there is a unique property to this action, which Leibniz found through his infinitesimal calculus. The exponential curve is the curve whose subtangent is always constant. (See animated figure 5d.)

In other words, the exponential curve is the curve whose characteristic of change is the same as itself!

This discovery highlights a crucial distinction between Leibniz’s method and Huygens’. With respect to the involute and evolute, the cycloid is the curve whose change is the same as itself. But from the more general method of Leibniz, it is the exponential curve that exhibits a characteristic of self-similarity. The importance of this distinction is underscored by Leibniz’s discovery of the relationship between the exponential curve and the catenary, which highlights the fact that the catenary expresses, more universally, the principle of least-action, not the cycloid.

As a result of this investigation, Leibniz discovered an entirely new type of transcendental function. He realized that even though every exponential had constant subtangents, the absolute length varied with the constant of proportionality. Leibniz found the existence of a new number, which he called, “b”, which forms the exponential curve whose subtangent is equal to unity. (See figure 5e.)

(Euler later changed the name of this number to “e”, whose historically misleading and quasi-blasphemous moniker it still wears today.)

With this new power to investigate physical action as the effect of a principle of change, new characteristics are brought to the surface that otherwise had remained hidden. For example, when the apparently uniform circular action is investigated from the standpoint of change of its subtangent, the existence of two discontinuities emerge, that otherwise were not visible. This is where the subtangent becomes infinite, which correspond, in Gauss’s idea, to the \/-1 and -\/-1. (See animated figure 5f.)