The “Earliest Uses” site suggests that the expression “uniform distribution” first appeared in Uspensky (1937), and “uniformly distributed” in Sakamoto (1943). Is that true?
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asked Aug 30 at 14:27
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2$\begingroup$ MO has a bit of a different philosophy towards self-answers than most of SE. This meta thread is from 11 years ago but my impression is that many still hold such a view. $\endgroup$– WojowuCommented Aug 30 at 14:48
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6$\begingroup$ In my view the well-researched self-answer is fine and valuable, and obviously not a case of self-promotion. On the other hand, this question and answer might fit even better on HSM than on MO. $\endgroup$– GNiklaschCommented Aug 30 at 15:02
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6$\begingroup$ It is perhaps not as widely known as it should be that the creator of the Earliest Uses sites passed away a couple of years ago; see westpascomuseum.org/pascohistory/historicalinformation/… I was one of those fortunate enough to have him as a high school teacher! $\endgroup$– so-called friend DonCommented Aug 30 at 15:13
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$\begingroup$ Is the Earliest Uses site still actively maintained? If so, they would presumably like to be informed about the references below. $\endgroup$– Timothy ChowCommented Sep 1 at 11:01
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It’s far from true. For one thing, Crofton in his famous paper (1869, p. 198) speaks of points “distributed with uniform density over the plane” (my bold). Moreover he refers to prior discussion in The Educational Times, where indeed one finds:
- vol. 10, Crofton (1868, p. 89): “Consider the given area as covered with points uniformly distributed”.
- vol. 9, Watson (1868, p. 37): “when the points $\mathrm O$ are uniformly distributed over $mn$”.
- vol. 8, Woolhouse (1868, p. 41): “as the $n$ points are uniformly distributed over the total spherical area”.
- vol. 7, Godfray (1867, p. 67): “points uniformly distributed throughout space”.
- vol. 5, Crofton (1866, p. 101): “a series of points are uniformly distributed along these radii”.
(The discussion is also interesting in that Godfray (above & vol. 6, p. 72) clearly anticipates “Bertrand’s paradox”.)
For another, de Moivre’s hypothesis of “uniform decrements” in Annuities upon Lives (1725, p. 20) and The Doctrine of Chances (1756, p. 263) assumed, in effect, a uniform distribution of deaths over the age interval [12, 86]. He didn’t say “distribution” but many after him did:
- Chrystal (1889, p. 569): “assume that the distribution of deaths throughout each year is uniform”.
- King (1887, p. 3): “Uniform distribution of deaths” “the deaths in each year of age are uniformly distributed”.
- Zeuner (1869, p. 85): “wenn die verlebte Zeit gleichmässig vertheilt wäre”.
- Wittstein (1862, p. 69): “sie muss die Sterbefälle gleichmässig über das Jahr vertheilen”.
- Zillmer (1861, p. 7): “Nimmt man aber an, daß das Sterben sich gleichmässig über das Jahr vertheile”.
- Gray (1849, pp. 64, 69): “the deaths being here supposed uniformly distributed” “De Moivre's hypothesis, of a uniform distribution of the deaths”.
answered Aug 30 at 14:27
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Mr John Pond, Esq. in the Philosophical transactions of the Royal Society of London, 1806, part II, p431: "this quantity was very uniformly distributed though the intermediate arc".
Rev. Baden Powell (Oxford) in same source, 1838, part II, p262: "Thus, even supposing the molecules uniformly distributed in the two media or portions of space, it is evident that within this stratum they will not be uniformly arranged."
''Disorsi e dimonstarzioni matematiche intorno a due nuove scienze'' by Galileo Galilei, in 1730 English translation. It would be interesting to see the wording in the original.
Probably there is plenty more even older.
I saw a reference to Isaac Newton describing the ether as uniformly distributed throughout space, but I didn't look for a citation.
Here is a 1686 usage. I'm going to stop looking now.
answered Sep 1 at 3:53
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$\begingroup$ Ah, but are those probability distributions? Many more examples can indeed be found with ‘uniformly distributed’ matter, or electric or magnetic charge, or load on a beam, etc. — but I think the claim was specifically about distributions of probability. $\endgroup$ Commented Sep 1 at 6:32
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$\begingroup$ @FrancoisZiegler Maybe not, maybe so, it depends on how hard one tries to see a distribution in words that don't use probabilistic language. Your Crofton 1868 example has the same issue. $\endgroup$ Commented Sep 1 at 12:41
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$\begingroup$ I’d say it’s pretty clear-cut: in a distribution of mass or charge or caloric or load or... (as in 2605 that I didn’t retain) we have the stuff distributed; whereas words like random or chance or probability signify that we only imagine doing (infinitely many) trials. $\endgroup$ Commented Sep 1 at 15:39