handle a system of equations with a maximum of two unknowns. In the second half of the
20th century the method gained in popularity due to its property of being resistent (robust)
against outliers. Nowadays Boscovich adjustment method is usually referred to as an l,|-
adjustment, since the L -norm of a vector is t lit" sum of absolute values of its entries.
V.
The mclliod of least-squares
Adrien-Marie Legendre (1752-1833), a professor of mathematics at the École Militaire in
Paris, was appointed by the French Academy of Sciences as member of various committees
on astronomical and geodetic projects, among them the committee on the standardization of
weights and measures. Fhe committee proposed to define the meter as 10 times the length
ol the terrestrial meridian quadrant through Paris at mean sealevel. The arc-measurements
took place in the period 1792-1795 and were analyzed, among others', bv both Laplace and
Legendre. Legendre 's 1805 11 publication on the determination of the orbits of comets,
contains a nine-page appendix in which for the first time the method of least-squares is
described, together with tin application of the method to the arc measurements.
Legendre used a different equation then Boscovich. Boscovich used the equation
5 [a(l -e2jae2(l -e2 )sin2 (p]A(p
w ith A(p=\. The arcs used by Legendre were not of one degree. Moreover, Legendre used a
parametrization which differed front the one used bv Boscovich. Since Legendre had the
determination of the meter in mind, he parametrized his equation in the length of a one
degree arc at 45 degree latitude. To obtain his equation, substitute
o y(cpj ep2) latitude of midpoint are
d a{\-e2) jjae2(1-e2) length of one degree are at 45 degree latitude
sin2(y(<z>, +<32)) |(l-cos(^, <p2)\
A(p (pn, s sn
and use the approximations sin(A^t), ^e2 ,fd ja( l-e2)e2
<p]2 si2c
As a result, we obtain
,<r' sin <pn cos(«p, +<p2
This is one equation in two unknowns, d and Legendre understood that the observed
latitude differences would correlate in case the arcs were connected. I le therefore trans
formed the above equation of differences into an equivalent undifferenced form. This can be
achieved by introducing an appropriate additional equation with an additional unknown. As
a result, we obtain Legendre's linear svstem of equations as
<PI
fPm.
1 sn
jsintpn cos(^] +<Pi)
<Pl
1 0
0
2 sin <Plm cos(<Pm +(Pl)_
Since Legendre had four connected arcs 5) at his disposal, he had to solve 5 equations in 3
unknowns. In order to solve his overdetermined linear system of equations, Legendre proposed to
determine x such that the sum of the squares of the residuals is minimized. In vector-matrix form
{y-Ax)(y-Ax) - mitt
By setting the derivatives of this quadratic form equal to zero, he shows that the solution satis
fies the consistent system ol linear equations (nowadays referred to as 'the normal equations')
>1
slm
X
y a
10
