# 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.

(Image from tess-elation.co.uk/johannes-kepler.)
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.

• 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."

• – Martin Sleziak Jul 21 '14 at 9:45
• "Then called Mascheroni's Theorem"—then in 1928, I suppose? The sentence reads like it means then in 1672, but that wouldn't make sense. – Wildcard Sep 20 '17 at 2:39
• @Wild, yes, 1928. – Gerry Myerson Sep 20 '17 at 5:26
• Fermat's proof of FLT (not written down because it was too long to fit in the margin) still awaits rediscovery. /s – none Aug 18 at 22:13

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.

• (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
• – NAME_IN_CAPS Jul 18 '14 at 8:10
• 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
• @Lembik Some details about Gauss and FFT are given in Heidemann et. al.: Gauss and the History of Fast Fourier Transform (pdfs.semanticscholar.org/1790/…) – user4503 Mar 29 '18 at 7:04

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.

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.

• So did Hipparchus make an error with 310952 instead of 310954, or was he counting something different? – Toby Bartels Mar 29 '18 at 5:08
• @TobyBartels, Hipparchus might not have made the error; it could be merely a scribal error in the manuscripts of Plutarch. I saw this issue raised in a research paper that I'll have to dig up. – Mark S Oct 6 '18 at 0:24
• Sure, that makes sense; it would hard to check for transcription errors in something like this, since most readers would be in no position to verify the number for themselves. – Toby Bartels Oct 18 '18 at 21:54

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

Almost at the same time Aristarchus of Samos calculated the sizes of the Sun and Moon, as well as their distances from the Earth in terms of the 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" for almost $2000$ years.

• Both valid, although it feels funny to refer to them both as lost in the same breath. With Eratosthenes, the precision was lost, but not the basic result of a spherical Earth, and people knew roughly how big Earth was (even Columbus was correct within 25%). But with Aristarchus, it's not a matter of his precision (which was actually terrible, around 95% error, off by more than an order of magnitude); rather, the whole concept of heliocentrism was lost, and people were not even aware that Sun is larger than Earth at all! – Toby Bartels Mar 29 '18 at 5:29
• Eratosthenes’s methods were not lost; for instance, they were refined by al-Khwarizmi. en.wikipedia.org/wiki/History_of_geodesy – Matt F. Jan 27 at 4:20
• @MattF. Did they use Eratosthenes’s methods? – Alexey Ustinov Jan 27 at 7:52

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.

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.

• 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

In 1994 integration (!) was discovered by medical researchers. The article A Mathematical Model for the Determination of Total Area Under Glucose Tolerance and Other Metabolic Curves has very big ipact (google scholar gives 299 citations).

Abstract

OBJECTIVE To develop a mathematical model for the determination of total areas under curves from various metabolic studies.

RESEARCH DESIGN AND METHODS In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method Gess than ±0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin.

• And in what sense was Integration lost? – Lucia Jun 28 '17 at 6:18
• @Lucia It was lost for medical researchers until these magnificent discovery. – Alexey Ustinov Jun 28 '17 at 6:21
• +1 just for the laughs. It reminds me of books on Quantum Information Theory and Quantum Computing which spend an awful lot of time reinventing (badly) linear algebra over the complex numbers. – Pedro Lauridsen Ribeiro Feb 13 '18 at 22:50
• As I understand it, medical researchers actually had integration in the form of Riemann sums; what Tai rediscovered in 1994 was the Trapezoid Rule. (In both cases without an error estimate, so not really mathematically satisfactory.) This makes sense, since the Applied Calculus textbooks, that we use (at least in the USA) to teach premed students about integration, mention Riemann sums but not any more sophisticated methods of numerical integration. – Toby Bartels Mar 29 '18 at 5:37
• @TobyBartels I've seen a few times insulin profiles drawn by Russian endocrinologists. This profiles are step functions like this. Each insulin dose is characterized by duration (x-axis) and amount of insulin units (y-axis). So the total daily insulin is NOT an area under this graph but the sum of hights. And I understood this discovery as a step to the our usual (mathematical) understanding of integal. – Alexey Ustinov Mar 29 '18 at 8:04

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.

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.

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

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

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.

• 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
• Why was Soddy looking at this topic at all? (The published paper is only the poem, with no acknowledgements, and his other mathematical papers were published later.) Without any context for his looking at this, it’s not clear that he found it independently. – Matt F. Jan 27 at 4:33

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.

• 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
• Although perhaps last paragraph would take on a different complexion if a simpler proof is found – samerivertwice Nov 8 '17 at 17:25

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

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".

Low-density parity-check codes (LDPC) were invented by Robert Gallager in his PhD thesis (1960). LDPC codes were forgotten until his work was rediscovered in 1996. Similar to Gallager’s LDPC codes were reinvented by different communities at roughly the same time.

LDPC was not quite forgotten; I wrote a monograph on it, published by MIT Press in 1963, and that monograph is still in print (although never a big seller). I investigated the possibilities for its use in a few military and space applications in the 60's and 70's as best I remember, but solid state technology was still too primitive for it to be economically feasible.

I feel quite remiss for not even mentioning it in a pretty substantial chapter on error correction coding in my Information Theory and Reliable Communication text book in 1968, nor in my Principles of Digital Communication text book in 2008. It was a deliberate choice rather than forgetfulness in both cases. In 1968, I didn't foresee the technology becoming better so fast, and in 2008 it had become a highly specialized technique that I didn't feel was the most important thing for first year graduate students to focus on.

When it was rediscovered, I was interested in following what others were doing on it, but was more interested in the other research topics I was working on at the time. At that time, what was needed was people really interested in detailed implementation, standardization, and manufacturing, and somehow I felt that others could do that better than I. There were enough people who were still familiar with my earlier work to focus again on it when it was reinvented, so the newer ideas and the older quickly became merged.

Technology works in strange ways, with some people more focused on basic research, some (particularly at places like Lincoln Labs, JPL, Qualcomm, Motorola) focusing more on using basic research, some motivated toward starting companies, and some in building things at well established companies and research labs. It has always amazed me that there are enough people at the interfaces that good ideas often become used when the time is ripe.

• The puzzling thing about this example that I've never fully understood is that Gallager remained an active researcher during the period in question, and presumably did not forget about LDPC codes. Maybe what happened is that Gallager's attention had moved on to other topics and so he was effectively "not around" to point out that LDPC codes had the properties that others were looking for? – Timothy Chow Nov 14 '16 at 21:30
• @Timothy Chow Yes, interesting question. Can you ask Gallager? There is an e-mail on his homepage rle.mit.edu/rgallager – Alexey Ustinov Nov 15 '16 at 1:57
• @TimothyChow Did you end up writing to Gallager? – Zach Teitler Feb 11 '18 at 16:36
• @ZachTeitler : I did not ask Gallager back in 2016 but, prompted by your comment, I tried writing to him yesterday. Haven't heard back yet. – Timothy Chow Feb 13 '18 at 3:18
• @TimothyChow what did Gallager say? Honestly I do not think there is anything profound about LDPC codes except the utilization of MAP decoding which was difficult computationally before the 90's. – T.... Feb 13 '18 at 11:59

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.

• 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 '16 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.

In a series of 3 articles ($*$), Luigi Onofri studied the group of permutations $\mathrm{Sym}(I)$ of an infinite countable set $I$. He notably proved (in the third article, 1929) that this group has exactly 4 normal subgroups: $$\{1\}\subset\mathrm{Alt}(I)\subset\mathrm{Sym}_{\mathrm{fin}}(I)\subset\mathrm{Sym}(I)$$ where $\mathrm{Sym}_{\mathrm{fin}}(I)$ is the subgroup of finitely supported permutations ("substitutions operating on finitely many elements") and $\mathrm{Alt}(I)$ is its subgroup of index 2 of even finitely supported permutations.

See https://math.stackexchange.com/a/2645097/35400 for more details.

This now classical result was rediscovered by Schreier and Ulam (1933, ${*}{*}$) and is so far exclusively attributed to Schreier-Ulam [with the notable but isolated exception of (${*}{*}{*}_1$), 1956 and (${*}{*}{*}_2)$, 1962], often as Baer-Schreier-Ulam theorem, referring to a subsequent extension (1934) by Baer to a classification of normal subgroups of the symmetric group over an arbitrary infinite set.

($*$) L. Onofri. Teoria delle sostituzioni che operano su una infinità numerabile di elementi, Memorie I, II, III. Annali di Matematica Pura ed Applicata. Memoria I: vol. 4(1) 73-106, 1927; Memoria II: vol 5(1), 147-168, 1928; Memoria III: vol. 7(1), 103-130. (restricted Springerlink: Memoria I, Memoria II, Memoria III)

(${*}{*}$) J. Schreier, S. Ulam. Über die Permutationsgruppe der natürlichen Zahlenfolge. Studia Mathematica (1933) Vol. 4(1), p.134-141, 1933. (EUDML unrestricted access)

(${*}{*}{*}_1$) W. Scott. The infinite symmetric and alternating groups. Pages 1-22 in: W. Scott, C. Holmes, E. Walker, Contributions to the theory of groups, National Science Foundation Research Project on Geometry of Function Space, report no 5, NSF-G 1126, U. of Kansas, 1956. (Quite rare book, not on MathSciNet; I got it sent to my library from another one. Seems to have been reedited recently (Amazon Link); Publisher: Literary Licensing, LLC (March 30, 2013) ISBN-10: 1258647044; ISBN-13: 978-1258647049; Year 2013.)

(${*}{*}{*}_2$) C. Kent. Constructive analogues of the group of permutations of the natural numbers. Trans. Amer. Math. Soc. 104 1962 347–362. (unrestricted pdf access)

Added: at the opposite, there is another famous result, originally due to the same authors Jósef Schreier and Stanislaw Ulam, for which they are scarcely quoted, namely: they proved that the group of orientation-preserving self-homeomorphisms of the circle is a simple group (${*}{*}{*}{*}$). (Still Google Scholar detects a certain number of quotations, so this would not qualify as "forgotten".)

(${*}{*}{*}{*}$) J. Schreier; S. Ulam. Eine Bemerkung über die Gruppe der topologisehen Abbildungen der Kreislinie auf sich selbst. Studia math. 5, 155-159 (1934). (EUDML unrestricted access)

• I've detected another isolated quotation of Onofri's result, namely in G. Stoller, The convergence-preserving rearrangements of real infinite series. Pacific J. Math. 73 (1977), no. 1, 227-231. – YCor Jun 17 '18 at 10:50

## 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

• 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
• Was this an oversight by Davenport, or was Cauchy's result completely forgotten at that time? – YCor Feb 16 '18 at 20:14

The number of binary De Bruijn sequences of order $n$ was determined by Camille Flye Sainte-Marie in 1894, but the result was forgotten and rediscovered by De Bruijn in 1946. However, the practice of calling them De Bruijn sequences remains entrenched.

The work of Charles Riquier (1853-1929) and Maurice Janet (1888-1983) on the formal properties of systems of partial differential equations (published 1910 - 1929) fell out of sight until J.-F. Pommaret modernised and promoted it (in his 1978 book Systems of Partial Differential Equations and Lie Pseudogroups).

Ernest Vessiot (1865 - 1952) is remembered today, if at all, for his contributions to differential Galois theory. In 1924 he produced a vector field formulation of partial differential equations (dual to Cartan's Pfaffian formulation) which was forgotten until 1985 (see paper by E. Fackerell in R. Martini Geometric Aspects of the Einstein Equations and Integrable Systems, Lecture Notes in Physics 239, Springer). In 1939 Vessiot used this formulation to link Darboux integrable second order PDEs to Lie groups of dimension $\leq 3$. This was subsequently extended by P. Vassiliou in 1986.

The breadth-first search, a classical algorithm of Computational Graph Theory, has an interesting history of invention.

According to Wikipedia, it was invented in 1945 by Michael Burke and Konrad Zuse, in his (rejected) Ph.D. thesis on the Plankalkül programming language, but this was not published until 1972. It was then reinvented in 1959 by E. F. Moore, who used it to find the shortest path out of a maze, and discovered independently by C. Y. Lee as a wire routing algorithm (published 1961).

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.

• 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
• @Andrew - the line about Galois' work being overlooked comes from E.T. Bell's unreliable account. For an accurate version, refer Amir Alexander's book Duel At Dawn. – Phil Harmsworth Jun 28 '17 at 7:45

Leibniz proved that $$1-\frac13+\frac15-\frac17+\frac19-\frac{1}{11}+\cdots = \frac{\pi}{4},$$ in 1674. However this result was known 300 years before to the Kerala School (India) identified with Madhava (see e.g. Borwein 2014; pdf).

In 1976 E. Salamin and R. Brent obtained independently a formula for approximating the number $\pi$ which is quadratically convergent; that is, every term one adds doubles the number of correct digits of $\pi$. We now know that this formula had been already proved by Gauss. It appears on page 6 of handbook 6, "Short Essays from various fields of mathematics, begun in May 1809", but the formula was forgotten. One reason could be that it is not of practical use if one has to make the calculations by hand. For more information see the chapter 7 of the book "$\pi$ unleashed" by J. Arndt and C. Haenel (Springer).

• I don't know if the first example fits. It's not that Leibniz had any access to that information, culturally speaking. It would be like claiming that since Martians had proved the Riemann Hypothesis, that proof is known and was forgotten. – Asaf Karagila Feb 13 '18 at 22:43

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)

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.)

• Thanks for the clarification at the end, but what are $\mu$ and $\alpha$ in relation to $d$, $m$, and $f$? – Toby Bartels Jun 3 '18 at 8:24

In the 1920's Ernst Steinitz discovered a remarkable theorem which today is known as Steinitz's Theorem - A graph G is isomorphic to the edge-vertex graph of a 3-dimensional convex polyhedron if and only if the graph is planar and 3-connected.

Only in the 1960's did the importance of what Steinitz had accomplished become clear when Branko Grünbaum and Theodore Motzkin recast/rediscovered what Steinitz had done in modern graph theory theory language that the importance of this remarkable result came to be exploited.

I would also like to comment on the phenomenon of "lost mathematics" in general. Perhaps another light in which to view the issue is that the mathematics at issue has gone to "sleep." Sometimes this occurs because the work is written in a language that does not have many readers. Sometimes the issue is that it was written by a person whose work does not have many "followers" and who did not have a broad context in which to understand what had been accomplished. Finally, often a "thread" of mathematics goes to sleep because with the mathematical tools of the time the line of work involved has gone as far as researchers at that time were able to carry it. When the "sleeping" thread gets reawakened and looked at, sometimes new ideas and tools are available to carry an earlier line of work much further.

Korselt's criterion (1899) for Carmichael numbers was born before Carmichael numbers (1910). Carmichael gave the same criterion in his article Note on a new number theory function.