From modern observations of the motion of artificial satellites it has been found that for the Earth, A3 is not zero and, hence, that the Earth has a non-negligible pear-shape distortion. An interesting discussion of how the motion of satellites can be used to determine the coefficients An can be found in King-Hele (1972).

Details of this correspondence can be found in, for example, Sheynin (1977a). The development of probability theory in the eighteenth century is described in detail in, for example, Daston (1988) and Hald (1998).

statistics. In his Philosophical Essay on Probabilities (1814) Laplace emphasized the importance of the theory of probability to natural philosophy and gave a general description of his thoughts on how to determine an appropriate 'average' value from a series of astronomical observations so as to minimize the possible error.

The details of this work can be found in his 1774 Memoir on the Probability of the Causes of Events,101 in which Laplace considered three observations registering the time of an astronomical event as ti, t2, and t3, with ti < t2 < t3. First, he observed that it is more probable that a given observation deviates from the truth by 2 seconds than by 3 seconds, by 3 seconds than by 4 seconds, etc. The law by which this likelihood diminishes as the difference between the observation and the truth increases is unknown, however.102

To determine this law, Laplace considered probability as a function of error in the form of a density $ (x) (which means that the probability of the error lying between x and x + dx is $(x )dx ).Henotedthat $(x) shouldbe symmetric about x = 0, since it is equally likely that the error is positive or negative; $(x) should approach zero as x increases, since the probability that the error is infinitely large is infinitely small; and $(x) dx = 1, since the total probability of all possible events is, by definition, unity. Laplace argued that, not only must $ decrease as x increases, so must d$/dx, and in the absence of any reason for a different rate of decrease for these two functions, he proposed that $ a d$ /dx which, given the other conditions that $ had to satisfy, leads to

$(x) = 2 me-m|x where m is a positive constant. This error distribution is now called the 'Laplace distribution' or the 'double exponential distribution'.

Now Laplace was faced with a problem in inverse probability. The function $ enabled him to compute the probability that a particular set of observations is made given knowledge of the true time of the event, but what is required is the probability that the true time of the event lies in a given range given that the set of observations is t1, t2, and t3. Here he invoked his 'fundamental principle':

If an event can be produced by a number n of different causes, the probabilities of these causes given the event are to each other as the probabilities of the event given the causes, and the probability of the existence of each of these is equal to the

100 L' essai philosophique sur les probabilites. English translation in Laplace (1951).

La memoire sur la probabilite des causes par les evenements (see, for example, Sheynin (1977b) and Hald (1998). Stigler (1986) contains a complete translation.

102 Quoted from Stigler (1986).

probability of the event given that cause, divided by the sum of all the probabilities of the event given each of these causes.

In modern notation, with E representing the event, Ci, i = 1,.n, the (mutually exclusive and exhaustive) causes, and P (A | B) the probability of A given B, Laplace's principle is equivalent to the equations

the second equation following from the first, since J2i= 1 P (C, | E) = 1.104 Laplace assumed that the same principle held in the case of a set of observations of a continuously varying parameter, so he could conclude that the probability of the true time of an event lying between r and r + dr, given the observations t1, t2, and t3, was f (r)dr, where f(r) a 0(r - t{)^(r - t2)$(r - ¿3)-

What criterion should be used to determine r ? Laplace considered two possibilities. In the first place, we could find the time such that it is equally probable that the event happened before or after it (which Laplace called the 'mean of probability'), or we could seek the time that minimizes the sum of the errors to be incurred multiplied by their probabilities (which he called the 'mean of error' or 'astronomical mean'). The first choice corresponds to finding r, such that

-ro J r and the second to saying that the minimum of the function

-ro is at t = r. Laplace proved that these amount to the same thing (as can be seen readily by evaluating F'(t) and solving F'(r) = 0). A simple calculation shows that r = t2 + m ln (1 + ie-m(t2-t1) - |e-m(t3-t2)), and Laplace demonstrated that as m ^ 0, r ^ f(t1 + t2 + t3), i.e. to the arithmetic mean. Laplace's analysis thus suggested that the arithmetic mean, which

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