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.