method of least-squares may seem 'natural' for a modern student of adjustment theory, its
discovery evolved only slowly from earlier methods of combining redundant observations'3.
In this contribution we sketch the historical line of development of these adjustment
methods in the second half of the 18th centurv.
The method of selected points
It is convenient to cast the problem of combining redundant observations in terms of vectors
and matrices. Suppose we are given a set of linear equations of the form
y A x
where y is a vector of observations, .1 is a given matrix of full rank and x is the vector of
unknown parameters. This set of linear equations is said to be overdetermined when there
are more observations than unknowns, m>n. The problem is to combine the in observations
so that one can solve for the n unknown parameters. If we restrict ourselves to linear combi
nations of the observations, we can write the general solution in the following form
x B y with B =(LA)~L
and where matrix L is a suitably chosen matrix defining the linear combinations. Different
choices of L give different linear combinations and therefore different solutions. In modern
terminology, matrix B is called a left-inverse of A, since B times A equals the identity matrix.
Before 1750 a popular, albeit subjective, method of solving an overdetermined set of linear
equations was the method of selected points. It consists of choosing n out of the m observa
tions (referred to as the selected points) and using their equations to solve for x. If the choice
falls on the first n observations, the corresponding L matrix takes the form
L
where is the identity matrix, lor the method of selected points, n residuals (the difference
between the observed and adjusted observations) are by definition equal to zero. Many scien
tists using this method calculated the remaining m-n residuals and studied their sign and size
to get an impression of the goodness of fit between observations and the proposed law.
The method is subjective because no clear rule is given which observations to select and
which to throw out. Selecting another set of n observations leads to a different solution for x.
Although the disadvantage of not using all observations was recognized, no simple method
existed to tackle this shortcoming. Sometimes all possible combinations of n observations
were considered and then averaged to obtain the final result. But since this approach requires
ïg m over n combinations, it was only practical for problems of low dimensions.
III. The method of averages
Tobias Mayer (1723-1762), professor of mathematics and head of the Göttingen observa
tory, made numerous observations of the moon with the purpose of determining the charac
teristics of the moon's orbit. In 1750 Mayer proposed a new method for adjusting his moon
data, a method which solved the above mentioned pitfall of the method of selected points.
Apart front his adjustment method, Mayer is also known for his other contributions to surve
ying and navigation. In 1752 lie invented the Repeating or Reflecting Circle, an instrument
for observing the angle between two celestial bodies. The accuracy of Mayer's instrument
was comparable to John Hadley's reflecting octant (1731), but had the advantage that it
could be used to measure angles of over 90 degrees'3. Mayer also contributed to solving the
mariner's 'longitude problem'. It was the British Parliament, which in 1714, offered the
mxl mxnnxl
/ixl nxm mxnxm nxn nxm
I 0
nxn nx(m-n)
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