This is my first time posting on HSM, so please bear with me if it's off-topic. I can move it to Stats.SE or Mathematics.SE if necessary.
A widely cited 1966 paper (with currently 1030 citations) mentions the "law of propagation of error" but does not actually state it or give any citation directly.
That 1966 paper does open though, with a citation to a 1939 paper by Raymond Birge which never mentions the "law of propagation of error" but Birge did write a 1932 paper which refers to what he calls "the well-known law of propagation of error", but again with no statement of the law, or citation to where that term was used previously.
I assume that the "law" to which they are referring is this one:
which is quite ubiquitous, and is given for example in this Wikipedia article. Surprisingly though, the entire Wikipedia article does not discuss the origins of this formula. Furthermore, if I search "law of propagation of error" on Google, I basically only find the above papers over and over again, which is quite frustrating. Both of Raymond Birge's papers heavily cite a 1894 book called "Method of Least Squares" of which I was able to find the 1910 edition.
A similar question has been asked here at HSM.SE but it was about when this formula became "prominent" not about when the "law" was first stated: When did error propagation become prominent in physics?
What is the origin of the "Law of propoagation of error"?
Update: The 1939 paper by Raymond Birge does actually state the "law" and it is indeed what I thought it was:
The reason I didn't find it earlier was because Adobe's optical character recognition was unable to find any instances of "law" or "propagation" in the text.