Between 1818 and 1832, Carl Gauss undertook the primary responsibility for making a geodetic survey of the Kingdom of Hannover. The task was exemplary of a great project. Its execution presented major technological difficulties that could only be overcome by developing new technologies based on new scientific principles, and its success would lay the basis for a transformation, through economic development, of the physical universe. But, perhaps even more important to Gauss, was that it provided an opportunity for him to, once again, do science, that is, prove that what everyone was thinking about the universe was wrong.

In the course of his survey, Gauss conducted the following crucial experiment: He measured the angular height of the pole star over the horizon from his observatory in Goettingen. His collaborator, Schumacher, measured the angular height of the pole star from his observatory in Altona, which was on the same meridian as Goettingen. From the difference of these angular measurements, Gauss calculated the distance along the surface of the Earth between the two observatories.

Then, with great effort, Gauss created a triangular grid over the entire Kingdom of Hannover. From these triangles, he made a second calculation of the distance along the surface of the Earth between the two observatories. The difference between the two calculations was 16" of an arc. A small error, by Baby Boomer standards, especially when compared to the Financial Times report yesterday that the "consensus" among financial experts concerning the prospects for tech stocks in 2001 was revised from 37% up to 30% down, without blushing. (That's why Baby Boomers like consensus. Everyone agrees to change their opinion together, so no one will be embarrassed when the consensus is proven wrong.)

However, for Gauss, this 16" of an arc discrepancy, like Kepler's 8' of an arc, was the opportunity to demonstrate that the way people were thinking was wrong. Not, that people were thinking SOMETHING wrong, but that the WAY they were thinking was wrong. (It is a characteristic of genius to be able to recognize when such small discrepancies are matters of principle, and not simply errors.)

What Gauss proved by this 16" of arc, as did Kepler, Fermat, and Leibniz previously, was that the mind must be free from any a priori, or "ivory tower" set of assumptions, such as those axioms, postulates and definitions of Euclidean geometry. Not, simply free from the particular axioms, postulates and definitions of Euclid, but free from any a priori set of assumptions.

Look back over the above described experiment, and discover the assumptions. First, the angular height of the pole star, is measured from the horizon. But, what is the horizon? It is not mathematically determined, but, it is physically determined as the perpendicular to the pull of gravity, as measured by plane leveller or plumb bob. A horizon defined in this way, will be tangent to the surface of the Earth at the point of measurement. Thus, the direction of the pull of gravity, is itself a function of the shape of the Earth. To calculate the distance along the surface of the Earth, from this angular change, one has to make an assumption about what is the shape of the Earth; i.e. if it's a sphere, the measurement is along a circle, if it's an ellipsoid, the measurement is along an ellipse. (Draw a circle and an ellipse. Draw tangents at different places. Draw perpendiculars to the points of tangency. What direction do these perpendiculars point? On a circle they all point to the center. On an ellipse they don't.)

Similarly, calculations from the triangular measurements, depend on what shape the triangles lie. (Compare triangles drawn on a sphere, an ellipsoid, or some irregular shape, like a watermellon.)

So, could some shape be found, on which the two different methods of measurement would agree?

Gauss rejected such "curve fitting" methods of thinking and made the revolutionary discovery that the shape of the Earth is that shape which is everywhere perpendicular to the pull of gravity, today called the "Geoid". The Geoid is not a geometrical shape, but rather a physically determined one. And, such a shape is not only non-uniform, but it is irregularly non-uniform, even changing over time.

Such an irregular, non-uniform surface, was, as Kepler's orbits, or the Fermat's path of least-time, or Leibniz' and Bernoulli's catenary, physically demonstrable, but unrecognizable by the generally accepted mathematics of the day. Having already been provoked by his teacher, Abraham Kaestner, Gauss had long before ceased to let mathematics dictate his thinking. Rather, he, like Leibniz invented a new mathematics. This extension of Leibniz' calculus did not rely on a priori assumptions about shape, but was a mathematics of transformations. Just as Leibniz' calculus made position along a curve a function of change, Gauss, and later Riemann, made shape a function of transformations, and curves a function of shape. The change that determines position along a curve, is itself determined by the transformation that generated the surface. This new mathematics required a new type of number; complex numbers. (The next several installments will work through these concepts in more detail.)

Gauss' method of inventing mathematics is rooted in Cusa's (of whom Kaestner placed great importance in his "History of Mathematics"). In "De Ludo Golbi", Cusa writes:

"And the soul invents branches of learning e.g. arithmetic, geometry, music, and astronomy and it experiences that they are enfolded in its power; for they are invented, and unfolded, by men....For only in the rational soul and in its power are the mathematical branches of learning enfolded; and only by its power are they unfolded. [This fact is true] to such an extent that if the rational soul were not to exist, then those branches of learning could not at all exist...."

"...the soul's reason, i.e. its distinguishing power, is present in number, which is from our mind, and in order that you may better know that that distinguishing power is said to be composed of the same and different, and of one thing and another thing just as is number, because number is number by virtue of our mind's distinguishing. And the mind's numbering is its replicating and repeating the common one, i.e. is its discerning the one in the many and the many in the one and its distinguishing one thing from another. Pythagoras, noting that no knowledge of anything can be had except through distinguishing, philosophized by means of number, I do not think that anyone else has attained a more reasonable mode of philosophizing. Because Plato imitated this mode, he is rightly held to be great."