I think is well known that greek scientists and even founding fathers of modern science did not use error propagation in their calculations. Today, instead, is unacceptable to work out any prediction of a physical theory without giving an estimate of how errors on the input parameters propagate into the computed quantity. So my question is:

  • when the use of error propagation became prominent? Who has been the pioneer who brought this into physics common practices?

4 Answers 4


I came across this question while trying to figure out when the "law of propagation of error" was first stated, which resulted in this question: When was the "Law of Propagation of Error" first stated? which now contains a reference to your question.

Your question is about when error propagation become prominent (rather than about when it first appeared). That part I can answer by saying that Raymond Birge in 1939 said that the question of how to assign an uncertainty has been discussed for decades but the subject matter of error propagation is one for which "many scientists still fail to avail themselves" and that "others frequently use the theory [of error propagation] incorrectly and thus arrive at quite misleading conclusions".

Raymond Birge was a prominent enough physicist, as he was the head of Berkeley's Physics Department and was regularly in contact with people like Gilbert Lewis, Robert Oppenheimer, and various other famous Nobel laureates or future Nobel Laureates. Therefore if there was people widely doing error propagation in a way that was "correct" in his eyes, he would have known about it or had access to the people who could advise him of the truth on the matter. In fact it was his frustration with inconsistencies in how people made measurements and reported fundamental physics constants that lead him to publish a series of papers about error propagation. People making measurements of fundamental physics constants would have been elite physicists (having enough money to do such experiments, and having also enough education to do such experiments), so if they were doing error propagation wrong, you can imagine that they were also teaching it wrong to the next generation of physics student.

As my above linked question describes, the chronologically next paper which comes up most often is this 1966 paper which says in the first sentence of the abstract:

The "law of propagation of error" is a tool that physical scientists have conveniently and frequently used in their work for many years

So by 1966, error propagation seems to have been quite popular. Perhaps Birge's papers in the 1920s and 1930s were what made error propagation popular. I already mentioned that Birge was the head of Physics at Berkeley and therefore taught many students that became prominent future scientists and had influence on a lot of other people too, but if that argument is not convincing enough you can see that between 1930 and 1940 his paper "Probable values of the general physical constants" was cited in Physical Review or Reviews in Modern Physics papers by I.I. Rabi (from Rabi Oscillations), Marie Curie (who needs not further words), Harold Urey (from the Miller-Urey experiment), Irving Langmuir (the namesake of the ACS journal Langmuir), Robert Millikan (from the Millikan oil drop experiment), E.E. Witmer (from the Wigner-Witmer rules), Walther Gerlach (from the Stern-Gerlach experiment), and about 70 other papers between 1930 and 1940 alone: this I think is enough to say that Birge was influential in science and probably were his papers about error propagation in the 1920s and 1930s.

Birge himself said in the 1939 paper that the idea of error propagation was not new, and he frequently cited a 1894 book by Mansfield Merriman called "Method of least Squares" for which a 1910 version can be found here. So perhaps Merriman (a civil engineer) popularized error propagation enough for it to be noticed by Birge (but insufficiently enough for Birge to be complaining about most people either not being aware of the theory or using it wrong), and then Birge popularized it to many of the prominent or to-become-prominent scientists of his time.

Conclusion: The mathematics of error propagation was discussed earlier by people like Mansfield Merriman, but even in 1939 it was not well-known by many people and was used incorrectly by many people, but Raymond Birge published several papers in the 1920s and 1930s on the topic of uncertainties in measurements, which were cited by many of the top scientists of the time. By 1966 error propagation was considered to be quite common.


One of the first pioneers was Gauß. He needed it, like the least squares law, for his astronomical calculations. In German it is called Gaußsches Fehlerfortpflanzungsgesetz.


I find this question baffling. Why would anyone be interested in 'propagating errors?'! Surely the question ought to be on estimating errors? Experimental physicists are only interested in how errors propagate in order to estimate errors - it's not an end in itself, but a means to an end and even then, the end here is to establish the level of confidence we have in the predicted value.

As soon as people developed the notion of a calculation - lost in time - and as soon as they developed the notion of one calculation depending upon another - again lost in time - they will have noticed as a matter of fact that 'errors propagate'. The likelihood is this will have been in astronomy, or land measurement as these were the earliest of sciences.

Archimedes used the method of exhaustion to calculate the area of a circle by an inscribed polygon and also provided an error estimate.


A few musings (at least one of which, I think, qualifies this to be an answer rather than a comment):

  • Strictly speaking, the quantities about which OP is asking are "uncertainties", not "errors" ("uncertainty" means the standard deviation of the forward probability distribution of a measurement given a fixed true value of the measurand, "error" means the difference between a single sample from that probability distribution and the true value - see the BIPM Guide to the expression of uncertainty in measurement. However, I realize this piece of pedantry is unhelpful in a question about the early history of the concepts, because those quantities were often called random errors in older publications.
  • The formal justification for propagating uncertainties through a calculation by multiplying by the modulus of the relevant partial derivative, and for combining uncertainties by adding in quadrature, is that these procedures emerge from the leading-order Laplace's method approximation to the integral that defines the standard deviation of the forward probability distribution. Hence, the procedures can't (or at least, shouldn't) have become prominent before Laplace's method was devised: according to the Wikipedia article on Laplace's method, that was in 1774.
  • The earliest example I've been able to find of someone distinguishing clearly between systematic errors and random errors in measurements is by Evelyn (1798, Philos. Trans. R. Soc. Lond. 88:133-182). Again, it wouldn't have made any sense for anyone to try to propagate uncertainties before that distinction was made.
  • The familiar (to us) formula for propagation and combination of uncertainties is stated and derived in section 149 of Galloway (1839, A treatise on probability, Adam and Charles Black). That source appears to be a republication as a book of an article from the 7th edition of Encyclopaedia Britannica, which was published in 1827.
  • $\begingroup$ This question asks when the method became "popular". Would you consider an attempt at answering my question, which is closer in spirit to your answer: hsm.stackexchange.com/q/12200/8052 ? $\endgroup$ Commented Aug 30, 2020 at 19:55
  • $\begingroup$ @user1271772 Consider it done. $\endgroup$ Commented Aug 30, 2020 at 20:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.