Why assume probability is the square of error? I'm an amateur trying to grasp Maxwell's original derivation of gas particle velocity probabilities. This led me to Herschel's review of Quetelet, which Maxwell may or may not have been directly inspired by. Herschel's informal description got me over the hump, mostly. I can now see how randomness leads directly to the need for the exponential function.
However, I'm less sanguine about the step in which Herschel declares that, to avoid negative numbers, the probability must be a function of the square of the error. Why not just use (e.g.) absolute value (which of course would lead to the wrong answer) instead of square?
My question is: is there some math/physics intuition I just don't have that makes the square the "obvious" choice in this situation, or is it that Herschel simply knew what the right answer was already via a more rigorous derivation and just reached for the simplest justification in this "dumbed down" explanation meant for the common man? Or, if that's too speculative to make a good question, then how about simply: why does it have to be the square instead of the absolute value?
 A: Finally I got around to reading the quaint English in "Herschel's review of Quetelet" mentioned in your link. @Phoenix87 has already answered your question: $|x|=\sqrt{x^2}$.
Also their interpretation of probability and statistics, as a procedure of arriving at best possible decisions (in the sense of being unbiased) in the face of our own uncertainty and ignorance, was heartwarming, because it is so close to E.T. Jaynes view of probability, which I really like. You may like to read entropy maximization procedure in E.T. Jaynes work on statistical mechanics/thermodynamics, or his book Probability theory: Logic of science.
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I know nothing of "Herschel's review of Quetelet". My answer to your question is on general grounds.
You may quantify error by either its absolute value, or its square, or any function that is positive and monotone increasing (such as for example, by even powers). In fact people have argued that, when dealing with error, taking absolute value is better than squaring (see Black Swan by N.N. Taleb, in which he argues against standard deviation which is nothing but square-root-mean of squared error), on the grounds that squaring operation gives more than proportionate weight to larger errors than smaller ones while absolute values do not face this problem. I think the ubiquitous use of squaring the error has to do with the ease it provides in subsequent mathematical analysis.
