Page 239 - Matrix Analysis & Applied Linear Algebra

234 Chapter 4 Vector Spaces

In September of 1801 Carl F. Gauss decided to take up the challenge of
finding this lost “planet.” Gauss allowed for the possibility of an elliptical or-
bit rather than constraining it to be circular—which was an assumption of the
others—and he proceeded to develop the method of least squares. By December
the task was completed, and Gauss informed the scientific community not only
where the lost “planet” was located, but he also predicted its position at fu-
ture times. They looked, and it was exactly where Gauss had predicted it would
be! The asteroid was named Ceres, and Gauss’s contribution was recognized by
naming another minor asteroid Gaussia.
This extraordinary feat of locating a tiny and distant heavenly body from
apparently insufficient data astounded the scientific community. Furthermore,
Gauss refused to reveal his methods, and there were those who even accused
him of sorcery. These events led directly to Gauss’s fame throughout the entire
European community, and they helped to establish his reputation as a mathe-
matical and scientific genius of the highest order.
Gauss waited until 1809, when he published his Theoria Motus Corporum
Coelestium In Sectionibus Conicis Solem Ambientium, to systematically develop
the theory of least squares and his methods of orbit calculation. This was in
keeping with Gauss’s philosophy to publish nothing but well-polished work of
lasting significance. When criticized for not revealing more motivational aspects
in his writings, Gauss remarked that architects of great cathedrals do not obscure
the beauty of their work by leaving the scaffolds in place after the construction
has been completed. Gauss’s theory of least squares approximation has indeed
proven to be a great mathematical cathedral of lasting beauty and significance.
Exercises for section 4.6

4.6.1. Hooke’s law says that the displacement y of an ideal spring is propor-
tional to the force x that is applied—i.e., y = kx for some constant k.
Consider a spring in which k is unknown. Various masses are attached,
and the resulting displacements shown in Figure 4.6.6 are observed. Us-
ing these observations, determine the least squares estimate for k.
x (lb) y (in)

5 11.1
7 15.4
y
8 17.5
10 22.0
12 26.3
x
Figure 4.6.6