Shaping the Earth - The Science of Navigation - page 50

of the newly established kingdom of Hanover in northern Germany. He
employed triangulation (using 26 triangles) and invented the method of
least squares to minimize the observation errors. Use of the least squares
method—of which, more anon—quickly became standard surveying prac-
tice. It provides the best-fit solution to a system of equations with more
measurements than unknowns—a situation that applies to surveying and
to many other fields that involve observation or experiment. Gauss's inter-
ests were so broad, however, that a rumor arose concerning his motivation
for the survey. Mathematicians had just discovered the possibility of other
types of space, weird curved spaces that later would be taken up by Ein-
stein. In fact this is just a myth: Gauss knew of the possibility of such non-
Euclidean spaces but never doubted that the space he surveyed was that of
Euclid. 17 The contributions of Gauss and others (such as Euler, Lagrange,
and Fourier) during the eighteenth and nineteenth centuries provided us
with most of the mathematical tools that are used in geodesy today.
Geodesy in the Electronic Age
Modern technology has changed the way that we do geodesy in the twenty-
first century. Indeed, modern technology has led to a redefinition of the
scope and applications of geodesy. Lasers are used to estimate the distance
between A and B with very high accuracy, whether these two points define
a residential lot boundary or the nearest points of the earth and the moon
(fig. 2.10). Radar altimeters and, more recently, satellite-borne interfero-
metric radars allow us to detect distances, and the slow change of distances
with time, to within a fraction of a centimeter, as we saw in figure 1.11.
Satellites and computers greatly assist our employment of these high-tech
measuring sticks, so that today we can measure and remeasure the entire
surface of the earth every few minutes, every day, continuously. Such
unprecedented monitoring capability—it is like a giant stethoscope mea-
suring the breathing and the heartbeat of an enormous beast—has shown
us that the surface of the earth is not stationary. It moves cyclically (due to
17. Euclidean space is the space we all learned about in school, in which the angles of
triangles add up to 180\. In non-Euclidean spaces, the angle can add up to more than 180\
(think of a triangle drawn on the surface of a sphere) or less (a triangle drawn on a saddle).
Gauss's geodesic triangle added up to 180\ with an error of less than two-thirds of a second
of arc. See Breitenberger (1984). For more on the history of the least-squares method, see
Chambert (1999, chap. 9) and Sheynin (2004).