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%Draft entry for C.C.Heyde and E.Seneta "Statisticians of the
%Centuries".
\noindent{\bf Rogerius Josephus BOSCOVICH }\\
b. 18 May 1711 - d. 13 February 1787
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\noindent{\bf Summary.} Rogerius Josephus Boscovich was a Jesuit priest from Dubrovnik,
Dalmatia, who developed in 1760 a simple geometrical method of fitting a
straight line to a set of observations on two variables using (constrained)
least sum of absolute deviations. This procedure was subsequently
popularised in
an algebraic form by Laplace.
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Rogerius Josephus Boscovich (Italian: Ruggiero Giuseppe
Boscovich; Croatian:
Rugjer (or Rudjer) Josip Bo\v {s}kovi\'{c}) was born in Ragusa
(now
Dubrovnik, Croatia), the sixth son of a merchant's agent. He was
educated at
the (Jesuit) {\it Collegium Ragusinum} in Ragusa and at the
(Jesuit) {\it %
Collegium Romanum} in Rome (Italy), but long before he had
completed his
training as a Jesuit priest he discovered his vocation as a
teacher of
mathematics. Indeed, in 1740 he succeeded his own teacher as
professor of
mathematics at the {\it Collegium Romanum}. In the years
following 1743
Boscovich was employed by Pope Benedict XIV as an adviser on a
variety of
mathematical problems. In particular, in 1750-1752 Boscovich and
the English
Jesuit Christopher Maire were commissioned to measure an arc of
the meridian
in the vicinity of Rome and to prepare a new geographical map of
the Papal
States. A detailed account of this geodetic survey was published
by Maire
and Boscovich in 1755 and a summary by Boscovich alone in 1757.
The Seven Years War of 1756-1763 was fought between Britain,
Hanover and
Prussia on one side and Austria and France on the other, and
resulted in the
loss of the French territories of Qu\'{e}bec and Maine in North
America.
This war brought Boscovich's skills as a diplomat into play. In
1758 he was
sent to Vienna (Austria) and in 1760 to Paris (France) and London
(England).
During his visit to London, Boscovich finalised his notes for the
second
volume of the poem in Latin hexameters by his fellow Ragusan
Benedikt Stay.
He also met many leading members of the Royal Society of London
and took the
opportunity of proposing a geodetic expedition to North America.
Later, in
1767, Boscovich was invited to participate in the 1769
astronomical
expedition to California, but his invitation was withdrawn when
the
organisers discovered that Jesuits were strictly excluded from
Spanish
territory in the Americas.
The Jesuit Order was suppressed throughout the world in 1773.
however
Boscovich managed to secure a pension from King Louis XV of
France which
lasted until his death in Milan (Italy) in 1787.
Boscovich made significant contributions to a wide range of
scientific
disciplines, his principal contribution being the first modern
treatise on
the atomic theory of matter. For detailed discussions of his
scientific
work, see Bursill-Hall (1994) and Whyte (1961). Here we shall be
concerned
with Boscovich's development of the
first {\it objective} procedure for fitting a linear relationship
to a set
of observations. [The earlier procedure of Tobias Mayer
is not
objective as the classification of the observations into groups
clearly
depends on the criterion employed in their classification.]
The fundamental mathematical problem which Boscovich addressed
in his
section of Maire and Boscovich (1755, pp. 497-503; 1770,
pp. 479-484), in
Boscovich (1757, pp. 391-392; 1961, pp. 88-93) and in his notes to
Stay (1760,
pp. 420-425; 1770, pp. 501-510) was that of determining the values
of the
coefficients $a$ and $b$ which best fit $n$ equations of the form
\[
y_i=a+bx_i\quad i=1,2,\ldots ,n
\]
The problem explicitly addressed by Boscovich in his 1755 and
1757 studies
concerned his attempt to reconcile the ten distinct values $%
b_{ij}=(y_i-y_j)/(x_i-x_j)$ which he had obtained for the slope
coefficient
by solving the $n=5$ equations two at a time. In 1755 Boscovich
had no firm
idea of what to do with these ten values. He first took an
unweighted
average of all ten values, and then an average of the eight
values which
remained after he had deleted the two with smallest denominators.
However,
in 1757 he formulated the principle that the values of $a$ and
$b$ should be
chosen in such a way that the errors
\[
e_i=y_i-a-bx_i\quad i=1,2,\ldots ,n
\]
\noindent should sum to zero and have minimum absolute sum.
From the adding-up condition, $\sum e_i = 0$, we find that $a$
is related to
$b$ by $a = {\bar y} - b{\bar x}$ where $\bar x = \sum x_i/n$ and
$\bar y =
\sum y_i/n$. Substituting this expression into the optimality
condition, we
find that we have to minimise $\sum|(y_i - {\bar y}) - b(x_i -
{\bar x})|$,
that is, we have to find the weighted median of the ratios
$b_i^{*} = (y_i -
{\bar y})/(x_i - {\bar x})\;\;i = 1, 2, \ldots, n$.
Boscovich did not explain how this problem could be solved in his
1757
summary but merely gave the results of the calculations. However,
in his
commentary to Stay's poem he showed how the ratios may be
calculated and
arranged in decreasing order by means of a geometrical procedure.
This
geometrical procedure takes the form of a cartesian diagram with
axes $x = 0$
and $y = 0$ in which the $n$ points $(x_i, y_i) \;\;i = 1,
2,\ldots, n$ have
been plotted together with a movable line. This movable line
starts in the
vertical position $x = \bar x$ and rotates in a clockwise
direction about
the centroid $(\bar x, \bar y)$. As it moves, the line passes
through each
of the $n$ points in turn and identifies the successive ratios
in decreasing
order. The weighted median of these ratios, and hence the minimal
value of
the absolute error function $\sum |e_i|$, is identified by
evaluating the
cumulative sum of the weights $|x_i - {\bar x}| \;\; i = 1, 2,
\ldots, n$ in
the order determined by the moving line, and choosing the point
for which
this cumulative sum first equals or exceeds one-half of the sum
of all $n$
such terms.
Boscovich's use of geometrical terminology in this context was
disapprobated
by Laplace (q.v.) who translated his procedure into an algebraic
format in
1793. Nevertheless, it should be noted that Boscovich probably
obtained the
ideas embodied in his solution procedure from a geometrical
diagram of this
type. the restricted variant of this procedure discussed by
Boscovich and
Laplace was subsequently generalised to any number of explanatory
variables
(with or without the adding-up constraint) by Gauss (q.v.) in
1809. However,
Gauss seems to have made no further use of the procedure after
that date.
See Farebrother (1987) for an account of the later history of
this $L_1$
procedure (subsequently renamed the method of situation by
Laplace) and the
closely related minimax absolute error or $L_\infty $ procedure.
More
recently, the $L_1$ procedure has served as the basis of a wide
class of
robust fitting procedures.
Boscovich has been severely criticised by later writers for
choosing to
publish some of his original scientific work in the notes to
Stay's poem.
Even if we allow that he could write in Latin with ease, it seems
inconceivable that he should have chosen to devote so much of his
spare time
to writing a detailed commentary on a poem in Latin hexameters.
However,
this is precisely what he did choose to do. Further, it seems
reasonable to
conjecture that Boscovich would have thought that this manner of
spending
his leisure time compared very favourably with that employed by
his
wealthier contemporaries (not to mention present-day
statisticians).
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\begin{thebibliography}{3}
\bibitem{1} Boscovich, R.J. (1757). De literaria
expeditione per
pontificiam ditionem et synopsis amplioris operis..., {\it
Bononiensi
Scientiarum et Artum Instituto atque Academia Commentarii} {\bf
4}: 353-396.
Reprinted with a Croatian translation by N. Cubrani\'{c},
Institute of
Higher Geodesy, Zagreb, 1961.
\bibitem{2} Bursill-Hall, P. (Ed.) (1994). {\it R.J.
Boscovich: His
Life and Scientific Work}, Insituto della Encyclopedia Italiana,
Rome.
\bibitem{3} Farebrother, R.W. (1987). The historical
development of
the $L_1$ and $L_{\infty}$ estimation procedures 1793-1930, in
Y.Dodge, {\it %
Statistical Data Analysis Based on the $L_1$- Norm and Related
Methods},
North-Holland Publishing Company, Amsterdam, pp. 37-63.
\bibitem{4} Maire, C. and R.J. Boscovich (1755). {De
Litteraria
Expeditione per Pontificiam Ditionem,} N. and M. Palearini, Rome.
French
translation by Fr.Hugon, N.M. Tilliard, Paris, 1770.
\bibitem{5} Stay, B. (1760). Annotated by R.J.Boscovich,
Philosophiae
Recentioris... Versibus {\it Traditae..., Tomus II,} N. and M.
Palearini,
Rome. French translation of sections 385-397 included in the
French
translation of Maire and Boscovich (1755), pp. 501-506.
\bibitem{6} Whyte, L.L. (1961). {\it Roger Joseph
Boscovich,} George
Allen and Unwin Limited., London and Fordham University Press,
New York.
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\hfill{R.W. Farebrother}
\end{thebibliography}
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