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The first thing a contemporary student of physics learns is the measurement error. As far as I understand, the idea of imprecision was totally foreign to natural philosophers at least until the end of 18th century. In 20th century the notion of experimental error became standard, and on top of that, a source of profound ideas.

Is it known when, and by whom, and in which paper, an error margin of an observation was first published and discussed?

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Astronomers had to deal with experimental errors to parametrize their geometric models at least as early as Hipparchus, and possibly earlier. There are some techniques and ad hoc methods that can be seen in hindsight as dealing with them in Ptolemy's Almagest, he discusses interpolation, for example. Ptolemy's "massaging" of Hipparchus's data even became a point of controversy recently, he apparently passed certain interpolations for observed data, with accusations of fraud and plagiarism advanced by Newton and others, see When did plagiarism become a major misconduct in academia? Here is a more charitable Gingerich's description in The Trouble with Ptolemy:

"Ptolemy clearly understood the geometry and realized that by stretching the period of greatest elongation he could get the relative placements of Venus and the earth (for him, the sun) much closer to the ideal positions he needed. Such approximations are characteristic of our most insightful scientists, who see them as a way to tackle otherwise intractable problems... It is clear that he deliberately moved away from the exact time of the greatest elongations in order to get the specific geometry he required... As Ptolemy wrestled with errors of measurement without any error theory, he was repeatedly forced into compromises to reconcile discordant observations."

In Optics V.2 Ptolemy proposes a series of experiments to substantiate his claim that "the angles [of refraction] do bear a certain consistent quantitative relation to one another with respect to the normals", and presents tabulations of relevant data. However, according to Smith's Ptolemy and the Foundations of Ancient Mathematical Optics

"There is, in fact, a specific mathematical law implicit in Ptolemy's tabulations, but its proper formulation in algebraic terms would have been beyond Ptolemy given the limitations of mathematical notation in his day."

Be it as it may, Ptolemy's Optics inspired a tradition, Islamic authors wrote elaborations that included new experimental data, e.g. Ibn al-Haytham's Book of Optics (1021) and al-Farisi's Optics (c. 1320). However, the first person to thematize experimental errors explicitly might be Ibn al-Haytham's contemporary al-Biruni. His interests included minerology, mechanics and even what we would call sociology. He talked of "errors caused by the use of small instruments and errors made by human observers", and of analysis (qualitative) of multiple observations to arrive at a "common-sense single value for the constant sought", to get a "reliable estimate", even suggesting the arithmetic mean. This is similar, and more specific, than Bacon's later four idols of the mind. Rozhanskaya and Levinova write in Statics (see Rashed edited Encyclopedia Of The History Of Arabic Science):

"The phenomena of statics were studied by using the dynamic approach so that two trends – statics and dynamics – turned out to be inter-related within a single science, mechanics... Numerous fine experimental methods were developed for determining the specific weight, which were based, in particular, on the theory of balances and weighing. The classical works of al-Biruni and al-Khazini can by right be considered as the beginning of the application of experimental methods in medieval science."

Needless to say, experimentation and dealing with errors became widespread in 17th century Europe, and even earlier Copernicus, Tycho Brahe and Kepler were conscious of astronomical observation errors. But theory had to wait for some development of probability and statistics. Quantitative theory of observation errors only appears in Simpson's memoir of 1755, which discussed several possible error distributions, including uniform and triangular distribution. For the further story see When did statistics become an integral part of physics?

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In The Oxford Guide to the History of Physics and Astronomy, under the heading Error and the personal equation, Kathryn Olesko writes:

Since Greek times astronomers have recognized that observations were afflicted by errors, that results based on them might be only approximate, and that the quality of data varied. Astronomers in early modern Europe took the first steps toward giving reliable estimates of those errors. Johannes Kepler, who used Tycho Brahe's observations to derive the elliptical shape of planetary orbits, was probably the first to construct a correction term that assigned a magnitude to error, ...

That would seem to point to the answer being Kepler's Astronomia nova (1609).

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  • $\begingroup$ Well, at least for astronomers. $\endgroup$ – Carl Witthoft May 2 '18 at 12:20
  • $\begingroup$ @CarlWitthoft, I can interpret that comment in multiple ways. I assume that you mean something like "Maybe a non-astronomer got there first". However, given the source of the quote, I think that would have to be at the least "non-astronomer non-physicist", and what other branch of natural philosophy at the time would want to deal with error analysis? Alchemy? $\endgroup$ – Peter Taylor May 2 '18 at 12:35
  • $\begingroup$ I'm not sure what "natural philosophy" is intended to mean. Theorists and practitioners (engineers in modern terms) existed in any number of fields other than astronomy. Take boat-builders or mapmakers as two examples. $\endgroup$ – Carl Witthoft May 2 '18 at 14:37
  • $\begingroup$ @CarlWitthoft, in modern terminology natural philosophy would be science, but I used that term for consistency with the question. (Contemporanous usage of science in modern terminology would be knowledge). $\endgroup$ – Peter Taylor May 2 '18 at 14:55
  • $\begingroup$ In Astronomia Nova Kepler is concerned with discrepancies between epicyclic path of Mars and Tycho's data, which he trusted. In other words, the error he looks at seems to be not an observation but a prediction error, which he ultimately eliminated by replacing epicycles by ellipses. $\endgroup$ – Conifold May 3 '18 at 22:53
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First of all, the idea of experimental error was not "introduced" but was forced on scientists by the nature of experiments themselves. As soon as you start measuring something, you immediately see that the results differ. If you measure something several times and obtain slightly different results, it is natural to take the average, which scientists did routinely. Now, until 17th century, almost the only exact sciences (which involved precise measurements) were astronomy and geography/geodesy. (I am aware of only one description of a physical experiment with measurements from antiquity, and it is very likely that this was really a "thought experiment", that is it was never actually performed and the numbers were computed theoretically. I mean Ptolemy's table of refraction.)

Another matter is a mathematical theory of errors. It was developed in the end of 18th century independently by Legendre and Gauss (the method of least squares). Legendre was the first, but this theory is usually attributed to Gauss. Their main motivation was astronomy and geodesy. But the method quickly spread to other sciences.

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  • $\begingroup$ Can you provide citations for the first documented use of averaging to resolve measurement variation? I'll also note that taking an average recognizes that error exists but not how to evaluate the magnitude. $\endgroup$ – Carl Witthoft May 2 '18 at 14:35
  • $\begingroup$ Re "it is natural to take the average, which scientists did routinely", the relative merits of a single measurement vs the mean of multiple measurements was a subject of debate in the Royal Society (of London) as late as the mid-18th century. Re priority on least squares (although this is getting a bit off-topic): what evidence is there that Legendre developed it before Gauss used it in 1801 to predict Ceres' path? $\endgroup$ – Peter Taylor May 2 '18 at 14:53
  • $\begingroup$ @PeterTaylor: 1. On Legendre. Here is an English translation of his paper: york.ac.uk/depts/maths/histstat/legendre.pdf 2. On the Royal society discussions: unlike the Royal society, I do not like to discuss trivial matters. $\endgroup$ – Alexandre Eremenko May 3 '18 at 0:00
  • $\begingroup$ Archimedes did measure a lot (some of his theorems he first obtained experimentally). Oresme did. Galileo did. Pascal did. But the idea of imprecision never crossed their minds. Even Hooke did not discuss it in "as the extension so the force". Sorry for the late comment. And whoever downvoted your answer did a bad job. $\endgroup$ – user58697 May 5 '18 at 3:27

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