For some pre-history involving Ptolemy and al-Biruni, see How was the idea of observation error introduced? Wikipedia sketches developments in the 18th century that were the context of Gauss's work in its History of statistics article. In particular, it credits first "formal study of theory of errors" to Cotes' Opera Miscellanea (1722) and its application to Simpson's memoir (1755). More qualitative discussions can be found earlier, in Galileo's Two Systems and Boyle's account of the experiments establishing his law (1662). Batten in A brief history of error quotes him as writing:
"Now although we deny not, but that in our table some particulars do not so exactly answer to what our formerly mentioned hypothesis might perchance invite the reader to expect; yet the variations are not so considerable, but that they may probably enough be ascribed to some such want of exactness as in such nice experiments is scarce avoidable... In the meantime (to return to our last-mentioned experiments) besides that so little variation may be in great part imputed to the difficulty of making experiments of this nature exactly, and perhaps a good part of it to something of inequality in the cavity of the pipe, or even in the thickness of the glass."
The 18th century developments are surveyed by Stahl in The Evolution of the Normal Distribution. The aforementioned Cotes speaks of observation errors pointedly and recommends the mean as a way to counter them (a practice adopted already by Tycho):
"Let p be the place of some object defined by observation, q, r, s, the places of the
same object from subsequent observations. Let there also be weights P, Q, R, S
reciprocally proportional to the displacements which may arise from the errors
in the single observations, and which are given from the given limits of error;
and the weights P, Q, R, S are conceived as being placed at p, q, r, s, and their
center of gravity Z is found: I say the point Z is the most probable place of the
object, and may be safely had for its true place."
Boškovic in a 1755 work on the shape of the earth proposed minimizing the sum of absolute errors, which leads to taking the median instead (which was, arguably, also favored by Galileo).
Simpson started assigning probabilities to errors and experimented with several error curves, including uniform and triangular, to support the rule of mean. A more systematic search for "the" error curve was done by Laplace in 1774, although the distribution he arrived at was peculiar and far from the bell curve. Lagrange proposed parabolic distribution in 1776, and raised cosine and logarithmic ones in 1781. Daniel Bernoulli questioned the rule of mean as late as 1777 explicitly citing "reliability":
"In this way, if the several observations can be considered
as having, as it were, the same weight, the center of gravity is accepted as the
true position of the object under investigation. This rule agrees with that used in
the theory of probability when all errors of observation are considered equally
likely.
But is it right to hold that the several observations are of the same weight or
moment or equally prone to any and every error? Are errors of some degrees as
easy to make as others of as many minutes? Is there everywhere the same probability?
Such an assertion would be quite absurd, which is undoubtedly the reason
why astronomers prefer to reject completely observations which they judge to be
too wide of the truth, while retaining the rest and, indeed, assigning to them the
same reliability."
Gauss published a derivation of the bell curve in 1809 in Theoria Motus Corporum Celestium, but his priority on inventing the least squares is heavily disputed. The first publication was by Legendre in 1805, see Stigler, Gauss and the Invention of Least Squares.
