# When was the "Law of Propagation of Error" first stated?

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.

That formula was stated (albeit in a rather different notation) and derived in section 149 of Galloway (1839, A treatise on probability, Adam and Charles Black), of which Google Books has the full text available. That work appears to be a republication as a book of an article from the 7th edition of Encyclopaedia Britannica, which was published in 1827.

I can't be sure that that's the earliest appearance of it, but as I argued in my answer to the above-mentioned related question, it can't have appeared very much earlier, because the formula relies on a conceptual understanding of errors that was first clearly described in 1798, and an approximation method for integrals which was invented in 1774.

• +1. Our of curiosity, how did you find these? Also, there's other ways of deriving the formula and the conceptual understanding about which you speak might not be strictly mandatory to come up with this formula: we sometimes see formulas appearing well before the relevant theory is developed. Aug 30, 2020 at 20:29
• I already knew that the formula was justified by a Laplace's-method approximation to a standard deviation, so I tried the Google Scholar search '"standard deviation" "laplace"', date-constrained to 1798-1850 (because I also already knew about the date for the conceptual framework). What I saw let me know that some workers in that period used the name "probable error" for the uncertainty, so I changed the search terms to '"probable error" integral laplace', still date-constrained to 1798-1850. The book by Galloway was the 5th of 6 results. Aug 30, 2020 at 21:10

The law of propagation of error is discussed at length in Text-book on geodesy and least squares, prepared for the use of civil engineering students by Charles L. Crandall, 1907.

There is also a paper "The essentials of error theory for practical engineers" by L. B. Tuckerman, in the Nebraska Blue Print, v. 13 (1914), pp.61-84 that discusses the "The law of ''propagation of error,'' i. e. the law expressing the relation between the error of the result, $$f$$, and the errors . . .") This zinger appears on p.63, but the essay is largely on this subject.

It is hard to believe either the concept or name originated in 1907. Both works I mention above seem pretty sound-minded to me.

• +1. How did you find these? Aug 30, 2020 at 19:48
• Google books with a 1950 time cutoff; then Hathi Trust to see the works themselves. Aug 30, 2020 at 19:50
• Also see the terminology in 1880s...books.google.com/… Aug 30, 2020 at 20:06