lure, (p tlie geodetic latitude of the midpoint of llie arc, and A(p the latitude difference ol the
two arc endpoints. The meridian curvature and its expansion are given as
M a?~62 3/2 =a(l~e2){l+fe2 sin2 <p+---}
(l-e sin <p) 2
with e~=(a~-b~)/a~ the eccentricity, a and b the half lengths of the major and minor axis,
and a(J-e~) the length of a degree at the equator. Using only the first two terms in the
expansion, the length of a one-degree arc can be written as
x x,+sin2^x2 w'lh or(l e2and x2 jae2(l-e2)
This is one equation in two unknowns, x j and xo. The arc length s and geodetic latitude (p
are determined from astronomical and geodetic measurements, while .v j and xj contain the
unknown dimensions of the ellipsoid of revolution. Although a minimum of two arcs is
needed to solve for the two unknowns, it is preferable to use more than two arcs. As a result
one obtains the following system of linear equations
~x,
Jm.
1 sin2 (pm
This is the system of equations which formed the start ol Boscovich analysis. Note that the
/I-matrix becomes near rank defect, when all arcs are close to the same latitude, for an
accurate determination of the two unknowns, it is therefore preferable to choose arcs at
widely differing latitudes. From the data available, Boscovich choose five such arcs (/n=5).
In his first analysis, Boscovich used the method of selected points. He choose the two arcs with
the largest difference in latitude. Not satisfied with the result obtained (the 3 residuals were
considered too large), lie considers all possible pairs ol measured arcs. This gave linn 10 selected
points to solve, but again he is not satisfied with the results obtained. After having struggled for
some time on how to proceed, Boscovich finally formulates his new method of solution in 1757.
I le states that the parameters xand xo should be chosen in such a way that the residuals sum
up to zero and have minimum absolute sum. In formula form these two conditions read
]^(x, -Xj -x2 sin2 cpj) 0 and \s, - x, - x2 sin2 <p, min
The first condition (although not essential) was motivated by the assumed symmetry in the
error distribution, while the second was chosen to get the adjusted values "as close as
possible' to the observed ones. Boscovich gave a graphical algorithm for solving his problem,
but no analytical one. The analytical proof of the solution was first given by Laplace in
1793. Using his principle, Boscovich first determined the two parameters xl and x'J. and
from them the flattening as f=xy/3xj. Here he only used the first term of the expansion
f (a-b)l a \e2 +}e4 +±eb+---
Hie value obtained bv Boscovich equals /=l/2-K>. which was smaller than the Battening
predicted by Newton. Based on a rotational ellipsoid as an equilibrium figure for a homoge
neous, fluid, rotating Earth, Newton obtained the value7=1/230. Boscovich' value is however
larger than the value known today (International Association of Geodesy (1980):7=1/298.257).
Boscovich' method was the first adjustment method that started from the principle of minimi
zing a function of the residuals. I lowever, although the method is objective and uses all the
observations, it did not reach the same level of popularity as Mayer's method, l ite method,
being nonlinear, was difficult to apply, while the at that time available algorithm could only
S1
2
1 sin
_*2_
X
y a
ni in
/=l i=t
