"Longitude Prize to those w ho c
a use
lgitude at sea. To determine longitude of a sh
practicable' method for detern
p at sea, the mariner needs to know both his
local time and the time at some standard location. Local time was readily determined, hut
the determination of standard time at sea was more complicated. Mayer's detailed lunar
tables (1755) made it possible to translate the instrument readings into longitude positions.
The use of Mayer's lunar tables was later superseded by John I Iarrison's marine chronometer
1 l--t (1759). In recognition of their contributions, both men were awarded part of the
Longitude Prize', with the larger sum going to Harrison1'.
Mayer studied the libration of the moon bv observing the changing position of the crater
Manilius as seen from the Earth. I sing spherical geometry, he found a linearized relationship
between his observables and some location parameters of Manilius and the moon's pole. This
in inconsistent system of 27 linear equations in 3 unknown parameters
gave him
1 n12
Mayer proposed to divide the 27 equations into 3 groups of 9 each, to sum the equations
within each group, and to solve the resulting 3 equations in the 3 unknowns. For a general
set of/// equations in unknowns, this approach amounts to a separation of the equations
into n groups, followed by a groupwise summation. For example in case 2. the correspon
ding L matrix takes the form
L
0
where the eand c-> are row vectors having only 1 s as their entries. Since one may use
averages instead of sums, the method became later known as the method of averages.
Mayer's method of averages soon became popular. It used all observations and it was very
simple to apply. However, due to the lack of an objective criterion of how to group the obser
vations, the method was still a subjective one.
IV.
The method of leant absolute deviations
To determine the Ear
(Principia7, 1687), tl
to Peru, Lapland t
s flattening as predicted by Newton's theory of gravitation
French Academy of Sciences organized arc-measurement expeditions
Cape of Good Hope in the period 1735-1754. These expeditions
aroused the interest in other countries and in 1750 Pope
Benedict XIV commissioned, the Jesuit and professor of
mathematics, Roger Joseph Boscovich (171 1-1787) to perform
a similar geodetic survey near Rome, the results of which were
published in 1755. In a summary of this report, published in
1757'*. Boscovich formulated his new method, now known as
the method of least absolute deviations, and applied it to the
data of th(> French and Italian arc measurements.
In order to understand the equations used by Boscovich, we
first need to introduce some elements from ellipsoidal geodesy.
Figure /.- Latitude arc measure- l or short meridian arcs, the arc length s (see figure 1can be
merits along a meridian. written as s= \l((p)A(p. with M((p) the meridian radius of curva-
Tl
a\l
*1
*2
y m
a ml
a ml
14
