# Rediscovery of lost mathematics

Archimedes (ca. 287-212BC) described what are now known as the 13 Archimedean solids in a lost work, later mentioned by Pappus. But it awaited Kepler (1619) for the 13 semiregular polyhedra to be reconstructed.

So there is a sense in which a piece of mathematics was "lost" for 1800 years before it was "rediscovered."

Q. I am interested to learn of other instances of mathematical results or insights that were known to at least one person, were essentially correct, but were lost (or never known to any but that one person), and only rediscovered later.

1800 years is surely extreme, but 50 or even 20 years is a long time in the progress of modern mathematics.

Because I am interested in how loss/rediscovery might shed light on the inevitability of mathematical ideas, I would say that Ramanujan's Lost Notebook does not speak to the same issue, as the rediscovery required locating his lost "notebook" and interpreting it, as opposed to independent rediscovery of his formulas.

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Since you mentioned Archimedes... – Lucian Jul 18 '14 at 3:12
This question is not a million miles distant from mathoverflow.net/questions/66075/… – Gerry Myerson Jul 18 '14 at 6:29
Heegner is not an example IMO. No one "rediscovered" his work in the sense of re-doing it. His method was seen, in retrospect, to prove what Baker and Stark had proven in the interim by different means. Then his original work (at that point better understood) was built upon. – NAME_IN_CAPS Jul 18 '14 at 8:12
Of course tehre is always the possibility that we do not even know that someon's proof was lost inthe first place. Or if we only have information about the proof and not the lost proof itself,we don't know if there really was a proof that got lost in the first place (think FLT). – Hagen von Eitzen Jul 18 '14 at 16:00
A remarkable example: www-math.mit.edu/~rstan/papers/hip.pdf – Pietro Majer Jul 21 '14 at 17:53

Just today, I read in the July 2014 Bulletin of the American Math Society, in the Mathematical Perspectives piece by Gerald Alexanderson, that "Lorenzo Mascheroni ... in ... 1797, proved that any [straight-edge and compass] construction ... can be carried out by compass alone. And that is where the problem stood until 1928 when a student browsing in a rack of books in a Copenhagen bookshop found a small book by Georg Mohr, an obscure Danish mathematician. It was ... published in 1672. It contained a proof of what was then called Mascheroni's Theorem. The contents of this volume had remained totally unknown."

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– Martin Sleziak Jul 21 '14 at 9:45

The algorithm now known as the FFT, described in Cooley and Tukey's landmark 1969 paper, was known to Gauss and appears among his unpublished works around 1805. I have also read that Archimedes' discovery of (at least parts of) integral calculus was found in a Byzantine manuscript whose pages had been "recycled" - the precious work of Archimedes washed (imperfectly, thankfully) from the pages to be refilled with some Greek clerical mumbo-jumbo. This would put Archimedes almost 1900 years ahead of Newton and Leibniz in that particular discovery.

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(Upvoted because the FFT example is a good one.) Archimedes did not discover integral calculus! He had a 'method of exhaustion' to determine the area of a region bounded by a curve, which is similar to Riemann-Darboux integration. However, this is not 'integration' until coupled with Descartes' idea of representing algebraic functions as curves. – Adam P. Goucher Jul 18 '14 at 7:58
You have a problem with phrasing here: removing parentheses you say "Cooley and Tukey's paper was known to Gauss", which is impossible. You should say "The algorithm known as the FFT, described by Cooley and Tukey in 1969, was known to Gauss..." – hobbs Jul 20 '14 at 6:54
@hobbs That's easy to fix with an edit instead of a comment. – David Richerby Jul 20 '14 at 12:05
Is there a copy (or translation) of Gauss's writing on FFTs available? I have often heard this story but I would love to know what he did in detail. – Lembik Aug 4 '14 at 9:49

Bernhard Bolzano .... ( interesting reading ) Much of his work was unpublished until much later (for reasons see the link), thus remaining largely unknown. For example, a theorem of Weierstrass is now known as the "Bolzano-Weierstrass theorem", acknowledging that Bolzano had proved it previously. He anticipated Cantor and Dedekind in work on doing calculus without infinitesimals. His example of a continuous nowhere-differentiable function is in a manuscript from 1830, but only published in 1930.

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This example has quite a short duration compared to the others listed so far but I thought I might mention it anyway: Wolfgang Döblin achieved important results in stochastic calculus shortly before his death in 1940 during military service. His proof of Itō's formula (which would not be proved by Itō until 1944) was recorded in a sealed envelope which was not opened until 2000, and has led to that result's being renamed the Itō-Döblin theorem in some textbooks.

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I am not sure how much this can count as rediscovery. There is no stochastic integral in Doblin's paper, and hence no actual Ito formula. What Doblin does, is to give a probabilistic characterization of a diffusion in terms of a time-changed Brownian motion. On the other hand this can be considered as even more anticipating, as the connection of random time changes and stochastic integrals was only discovered in 1965, by Dubins-Schwarz and Dambis... – Stephan Sturm Jul 19 '14 at 2:34
A good overview of Doblin's work in the "pli chachete" is given in this article by Bernard Bru and Marc Yor: ams.org/mathscinet/search/… – Stephan Sturm Jul 19 '14 at 2:35
My knowledge of the topic is fairly shallow, so if you would like to edit this answer then please be my guest. – Ian Morris Jul 19 '14 at 16:54

The Schröder-Hipparchus numbers give an interesting example:

According to a line in Plutarch's Table Talk, Hipparchus showed that the number of "affirmative compound propositions" that can be made from ten simple propositions is 103049 and that the number of negative compound propositions that can be made from ten simple propositions is 310952. This statement went unexplained until 1994, when David Hough, a graduate student at George Washington University, observed that there are 103049 ways of inserting parentheses into a sequence of ten items. A similar explanation can be provided for the other number: it is very close to the average of the tenth and eleventh Schröder–Hipparchus numbers, 310954, and counts bracketings of ten terms together with a negative particle

The problem of counting parenthesizations was introduced to modern mathematics by Schröder (1870).

If this interpretation is correct it's a non-trivial combinatorial problem that Hipparchus solved. Hipparchus must have known some combinatorial techniques that are not in any contemporary sources that survive today but which were rediscovered in the 19th century.

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If "essentially correct" is not necessarily the same as "rigorously established", then Lucjan Emil Boettcher qualifies with his pioneering work in what is now known as holomorphic dynamics, 20 years prior to Pierre Fatou and Gaston Julia (without using the notion of normal family, though, which was not available when he was working on these topics), and writing in German, Polish and Russian. See Mathematicians whose works were criticized by contemporaries but became widely accepted later

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Grassmann is another good example from your link... – Steve Huntsman Jul 18 '14 at 4:05

The Rogers-Ramanujan identities have a similar story. They were discovered and proved by Leonard James Rogers in 1894 and then promptly forgotten. Ramanujan then discovered them in 1913 without a proof.

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The number theory work of Fermat might be an example. He was rather secretive about his methods and much has to be rediscovered later by Euler. This includes Fermat's two-square theorem: It was first mentioned by Fermat as a theorem in a 1640 letter to Mersenne and also analogous statements about primes numbers of the form $x^2+2y^2$ and $x^2+3y^2$ were made in a 1654 letter to Pascal. While Fermat claimed to have solid proofs, he did not write more than a very vague sketch using infinite descent. Euler first became aware of Fermat's work around 1730. It took Euler until 1749 to prove Fermat's two-square theorem and until 1772 to prove the analogous statements about primes numbers of the form $x^2+2y^2$ and $x^2+3y^2$.

A more knowledgeable person could certainly present more examples in the work of Fermat and Euler. I do not include "Fermat's last theorem" as it seems virtually impossible to me that Fermat possessed a correct proof for this.

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Yes, Weil's history of number theory is full of Euler working hard to prove Fermat's claims. – Marius Kempe Apr 12 '15 at 17:32

There is René Descartes' formula relating the radii of four mutually touching circles, which he sent to Princess Elisabeth of the Palatinate in 1643, rediscovered in 1826 by Jakob Steiner, in 1842 by Philip Beecroft, and again in 1936 by chemistry Nobel prize winner Frederick Soddy who announced his discovery in Nature in verse:

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.


It is curious, given Soddy's "kissing", that Beecroft published his result in The Lady's and Gentleman's Diary.

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Also cited in MO question, Mathematical research published in the form of poems. – Joseph O'Rourke Jul 18 '14 at 23:53
There's probably no name for this poetic form, but it employs roughly ballad meter (alternating 4-stress and 3-stress lines, but here some of the 3s are replaced by 4s) and a fun little 10-line stanza that's almost like a cut-down sonnet, with an abcb-dd-ee-fa rhyme. – hobbs Jul 20 '14 at 7:33
@hobbs: it reminds me nicely of Lewis Carroll’s long but charming Phantasmagoria, which also uses a mixed pattern of 4-foot and 3-foot lines, in that case in 5-line stanzas (4 / 3 / 4 / 4 / 3). – Peter LeFanu Lumsdaine Jul 21 '14 at 13:42

Eratosthenes calculated radius and circumference of Earth with accuracy $\approx 2\%$. Of cause it is "only applied" mathematics, but extremely advanced for that time. Next time the same accuracy was achieved only in 19th century.

Almost at the same time Aristarchus of Samos calculates the sizes of the Sun and Moon, as well as their distances from the Earth in terms of Earth's radius. His Heliocentrism was rejected until it was successfully revived by Copernicus, after which Johannes Kepler described planetary motions with greater accuracy, with Kepler's laws, and Isaac Newton gave a theoretical explanation based on laws of gravitational attraction and dynamics.

In both cases we have results which were "lost" almost for $2000$ years.

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The fact that any continuous self map of an interval with a point of period three must have periodic points of all periods was proved in the paper "Period Three Implies chaos" by Li and Yorke which was published in 1975. This was considered new at the time but was later seen to be a very special case of Sharkovskii's theorem (which gives a complete ordering of the natural numbers such that a point of period $n$ implies points of all periods following it in this ordering). Sharkovskii had published his theorem a decade earlier in 1964.

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There is a nice, slightly speculative, example in Conway and Doyle’s Division by Three paper:

In 1926 Lindenbaum and Tarski announced, in an infamous paper that contained statements (without proof) of 144 theorems of set theory, that Lindenbaum had found a proof of division by three. Their failure to give any hint of a proof must have frustrated Sierpiński, for it appears that twenty years later he still did not know how to divide by three. Finally, in 1949, in a paper ‘dedicated to Professor Wacław Sierpiński in celebration of his forty years as teacher and scholar’, Tarski published a proof. In this paper, Tarski explained that unfortunately he couldn’t remember how Lindenbaum’s proof had gone, except that it involved an argument like the one Sierpiński had used in dividing by two, and another lemma, due to Tarski, which we will describe below. Instead of Lindenbaum’s proof, he gave another.

We tried and tried and tried to adapt [Sierpiński’s] method to the case of dividing by three, but we kept getting stuck at the same point in the argument. So finally we decided to look at Tarski’s paper, and we saw that the lemma Tarski said Lindenbaum had used was precisely what we needed to get past the point we were stuck on! So now we had a proof of division by three that combined an argument like that Sierpiński used in dividing by two with an appeal to Tarski’s lemma, and we figured we must have hit upon an argument very much like that of Lindenbaum’s. This is the solution we will describe here: Lindenbaum’s argument, after 62 years.

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See also arxiv.org/abs/1504.01402 for a follow-up. – Emil Jeřábek Apr 12 '15 at 16:08

The Tonelli–Shanks algorithm for finding square roots modulo a prime number. Today this is well known, it is e.g. used in the quadratic sieve method to factor large numbers. Alberto Tonelli discovered the algorithm in 1891, it was re-discovered by Daniel Shanks in 1973.

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Tonelli–Shanks algorithm is a problem V.2 from Vinogradov's "Elements of Number Theory" (my 6th edition was published in 1953, 20 years before Shanks). Similar algorithm can be found in the Grave's "Elementary course of number theory (1913). Grave calls it Korkine's method. Korkine proposed (1909, postmortal article) method for solving $x^n=a(mod p)$, but he used primitive roots, see mathnet.ru/php/… – Alexey Ustinov Mar 8 at 9:32

There is also Redfield who discovered the cycle index series and anticipated combinatorial species in enumerative combinatorics. His first paper was published but ignored. His second paper was rejected for publication. The cycle index series was then rediscovered by Polya ten years later.

Redfield, J. Howard (1927). "The Theory of Group-Reduced Distributions". American Journal of Mathematics 49 (3): 433–455. doi:10.2307/2370675. JSTOR 2370675. MR 1506633.

G. Pólya (1937). "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen". Acta Mathematica 68 (1): 145–254. doi:10.1007/BF02546665.

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Branko Grunbaum wrote a paper called "Lectures on lost mathematics" where he mentioned various mathematical results, theories and problems that have been lost.

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Dehn's proof that the mapping class group is finitely generated by twists, in 1938, was independently discovered (and simplified) by Lickorish in 1962 and 1964. Dehn's coordinates for curves on surfaces, also in 1938, was rediscovered by Thurston (and greatly extended) in 1988.

In all cases I've give the publication dates; the relevant mathematics was being publicized earlier: at least 1922 for Dehn and 1976 for Thurston. But Dehn's work was interrupted by two world wars... See Stillwell's remarks in his collected translations "Papers on group theory and topology".

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## The Cauchy-Davenport Theorem

Let $t$ be a non-negative integer and let $x_1, ..., x_t$ be nonzero elements of $\mathbb{Z}_p$ which are not necessarily distinct. Then the number of elements of $\mathbb{Z}_p$ that can be written as the sum of some subset (possibly empty) of the $x_i$ is at least $\min\{p,t+1\}$. In particular, if $t\geq p-1$, then every element of $\mathbb{Z}_p$ can written in this way.

Davenport proved this result in 1935, which is used quite extensively in the circle method and the Waring's problem, without knowing that in fact this was a result proved by Cauchy in 1813.

Davenport, H, A historical note. J. London Math. Soc. 22, (1947). 100–101

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Yes, Cauchy-Davenport was mentioned in the question I link to in my comment on this question. – Gerry Myerson Aug 5 '14 at 23:41

One of the most famous examples of "lost mathematics" must surely be the whole subject of Galois theory and, to some extent, group theory as developed by Galois. Galois attempted to publish his work several times, but it was overlooked by the likes of Cauchy and Fourier (in his defence, Fourier died). Luckily nine years after his death Galois's papers found their way to Liouville and they were published.

Under a strict interpretation of the question, Galois theory is perhaps not an example of "lost mathematics" because it not recreated by some one else. On the other hand, it definitely was "lost" and it was then later "rediscovered" by Liouville.

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Part of Galois' work itself - the proof of impossibility of solving the quintic by radicals, can also be considered a piece of rediscovered lost mathematics, since Paolo Ruffini's essentially complete proof had been ignored (possibly because times were not ready to such a revolutionary idea as an impossibility result). – Pietro Majer Sep 5 '14 at 14:53

The formula of Moran for the dimension of self-similar sets (Moran (1946) Additive functions of intervals and Hausdorff measure, Proceedings of the Cambridge Philosophical Society, vol. 42) was rediscovered by Hutchinson (Hutchinson (1981) Fractals and self similarity. Indiana Univ. Math. J. 30 (5): 713–747)

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Cusick T., Flahive M. The Markoff and Lagrange spectra. AMS, 1989, page 2:

The result that $\mu(\alpha) \ge \sqrt{5}$ for all real $\alpha$ is often referred to as Hurwitz's Theorem, even though it is contained in Markoff's work. This is because Hurwitz [1891] proved $\mu(\alpha) \ge \sqrt{5}$ directly, whereas the paper of Markoff [1879] approaches the problem via quadratic forms. In fact, even earlier, Korkine and Zolotareff [1873, pp. 369-370] stated the result $\sqrt{d(f)}/m(f)\ge \sqrt{5}$ and also stated that the next largest value of $\sqrt{d(f)}/m(f)$ is $\sqrt{2}$. Markoff [1879] refers to their work as the starting point for his own.

(The minimum $m(f)$ of an indefinite binary quadratic form $f(x,y) = ax^2 + bxy + cy^2$ with real coefficients and positive discriminant $d(f) = b^2 — 4ac$ is defined by $m(f) = \inf|f(x,y)|$, where the infimum is taken over all pairs of integers $x$, $y$ not both zero.)

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