"How did Mayer address his overdetermined system of equations? His approach
was a simple and straightforward one, so simple and straightforward that a
twentieth-century reader might arrive at the very mistaken opinion that the
procedure was not remarkable at all. Mayer divided his equations into three
groups of nine equations each, added each of the three groups separately,
and solved the resulting three linear equations for \(\alpha\),
\(\beta\) and \(\alpha sin\theta\) (and then solved for \(\theta\)).
His choice of which equations belonged in which groups was based upon the
coefficients of \(\alpha\) and \(\alpha sin \theta\). The first group
consisted of the nine equations with the largest positive values for the
coefficient of a, namely, equations 1,2, 3, 6, 9, 10, 11,12, and 27. The
second group were those with the nine largest negative values for this
coefficient: equations 8, 18, 19, 21, 22, 23, 24, 25, and 26. The remaining
nine equations formed the third group, which he described as having the
largest values for the coefficient of \(\alpha \sin \theta\)."
Stigler (1986, p.21)
Stigler (1986) says:
"The development of the method of least squares was closely associated with
three of the major scientific problems of the eighteenth century: (1) to
determine and represent mathematically the motions of the moon; (2) to
account for an apparently secular (that is, nonperiodic) inequality that had
been observed in the motions of the planets Jupiter and Saturn; and (3) to
determine the shape or figure of the earth. These problems all involved
astronomical observations and the theory of gravitational attraction, and
they all presented intellectual challenges that engaged the attention of
many of the ablest mathematical scientists of the period.
Over the period from April 1748 to March 1749, Mayer made numerous
observations of the positions of several prominent lunar features; and in
his 1750 memoir he showed how these data could be used to determine various
characteristics of the moon's orbit. His method of handling the data was
novel, and it is well worth considering this method in detail, both for the
light it sheds on his pioneering, if limited, understanding of the problem
and because his approach was widely circulated in the major contemporary
treatise on astronomy, having signal influence upon later work."
His analysis led to this equation:
$$\beta - (90-h) = \alpha \sin(g-k) - \alpha \sin\theta \cos(g-k)$$
According to Stigler (1986, p. 21), this "equation would hold if no errors
were present. The modern tendency would be to write, say":
$$(h - 90) = - \beta + \alpha \sin (g - k) - \alpha \sin\theta \cos(g - k)
+ E$$
treating \((h - 90)\) as the dependent variable and \(-\beta\),
\(\alpha\), and \(-\alpha \sin \theta\) as the parameters in a linear
regression model.