equations that arise in these studies. Later volumes
studied the local forces acting between molecules and
the large-scale effect of these local interactions. He
applied this work to the study of pressure and density,
the nature of gravity, capillary action, sound, refrac-
tion, and the theory of heat.
In 1812 Laplace published his famous piece on the
topic of probability. Motivated by the analysis of errors
in observation, Laplace developed the theory of proba-
bility from beginning principles and made clear the
mathematical underpinnings of many aspects of the
topic. He provided his own definition of probability
and used it to justify the basic mathematical manipula-
tions. The work discusses his famous
LEAST SQUARES
METHOD
, the B
UFFON NEEDLE PROBLEM
, and B
AYES
’
S
THEOREM
. In analyzing the distribution of errors in sci-
entific observations, Laplace also applied this work to
the accurate determination of the masses of Jupiter, Sat-
urn, and Uranus, as well as to improving triangulation
methods in surveying, and correct determination of
longitude and latitude in geodesy.
Under Napoleon, Laplace became a count of the
empire in 1806 and was awarded the title of marquis in
1817. Laplace wrote that he believed that the nature of
the universe is completely deterministic, meaning that
the motions all objects in the solar system are predeter-
mined by the initial conditions at the start of the uni-
verse. When Napoleon asked him where God fit into
this view, Laplace is said to have replied: “I have no
need of that hypothesis.”
Laplace died in Paris, France, on March 5, 1827. It
is not possible to exaggerate the influence Laplace had
on the development of the mathematical theory of
mechanics. A number of fundamental concepts, such as
the Laplace operator in potential theory and the
Laplace transform in the study of differential equa-
tions, are named in his honor.
Latin square An n×nLatin square is an arrange-
ment of the first nLatin letters A, B, C, D, … in a
square array in such a manner that no row or column
of that array contains two identical letters. For exam-
ple, the arrangement below is a 3 ×3 Latin square:
ABC
CAB
BCA
A Latin square is called diagonal if, in addition, no let-
ter is repeated in either diagonal of the square. The fol-
lowing, for example, is a diagonal 4 ×4 Latin square:
ABCD
CDAB
DCBA
BADC
No 3 ×3 Latin square is diagonal.
There is just one 1 ×1 Latin square, two 2 ×2
Latin squares (neither of which are diagonal), 12 3 ×3
Latin squares, and 576 4 ×4 Latin squares. The num-
ber of Latin squares of a given order grows extremely
rapidly, as the following table shows:
There are more 15 ×15 Latin squares than there are
atoms in the universe (which physicists calculate to be
about 1081).
Two Latin squares of the same order are called
mutually orthogonal if all the pairs of letters that
appear when the two squares are superimposed are dif-
ferent. For example, the two squares shown below are
mutually orthogonal. Here we have used letters of the
Greek alphabet for the second square. The resultant
array of pairs of elements that appears is called a
Graeco-Latin square.
+ =
This square solves the famous “officer problem” first
posed by L
EONHARD
E
ULER
(1707–83):
AαBβCγDδ
BγAδDαCβ
CδDγAβBα
DβCαBδAγ
αβγ δ
γδαβ
δγβα
βαδγ
ABCD
BADC
CDAB
DCBA
nNumber of n×nLatin Squares
11
22
312
4 576
5 161,280
6 812,851,200
7 61,479,419,904,000
8 108,776,032,459,082,956,800
9 5,524,751,496,156,892,842,531,225,600
10 9,982,437,658,213,039,871,725,064,756,920,320,000
302 Latin square