What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple that it's surprising no one thought of it sooner.
The example that makes me ask is the 2011 paper John Baez mentioned called "Two semicircles fill half a circle", which proves a fairly simple geometrical fact similar to those that have been pondered for thousands of years.
This proof of the irrationality of $\sqrt 2$ appears to have been discovered in 1892 by A.P. Kiselev:
If $\sqrt2$ is rational, let $\triangle ABO$ be the smallest possible isosceles right triangle whose sides are integers.
Construct $CD$ perpendicular to $AO$ with $AC = AB$. $\triangle OCD$ is another isosceles right triangle.
$AC=AB$, therefore $AC$ is an integer, therefore $OC = OA - AC$ is an integer. $\triangle OCD$ is isosceles, so $OC=CD$ and CD is an integer. $CD$ and $BD$ are equal because they are tangent to the same circle, so $BD$ and $OD = BO-BD$ are integers. But then $\triangle OCD$ is an isosceles right triangle with integer sides, contradicting the assumption that $\triangle ABO$ was the smallest such.
I am amazed that this wasn't found by the Greeks, because it is so much more in their style than the proof that they did find.
(Source: http://www.cut-the-knot.org/proofs/sq_root.shtml#proof7)
This is not a theorem but a result which was discovered "too late".
The number $e$ hiding inside Pascal's triangle was found it by Harlan Brothers in 2012, in a very simple way!
Apéry's 1975 proof of the irrationality of of $\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}$ took the experts by surprise, and uses tools that had been available for a very long time.
See A Proof that Euler Missed for more details.
An example mentioned for Martin Gardner (Martin Gardner's New Mathematical Diversions from Scientific American): Morley's trisector theorem is elementary enough to be proved two millenia ago, but unknown until 1899.
Informally, the historical definition of an Archimedean solid requires that all its faces are regular polygons and that each vertex of the solid locally looks the same. Since the ancient Greeks it was common knowledge that up to scaling, rotating and reflecting, there are 13 types of Archimedean solids.
It was only in the 20th century that it was noticed that there is a 14th solid which complies with the above definition, the Elongated Square Gyrobicupola. It arises from the Rhombicuboctahedron (one of the 13 established Archimedean solids) by twisting one of its octagonal "caps" by 45 degrees. However, those two solids are not identical.
Since this solid doesn't look as symmetric as the "proper" Archimedean solids (its symmetry group doesn't act regularly on its vertices), nowadays the definition typically is sharpened.
The theorem that $$\text{PRIMES is in P}$$ was proven only in 2002 (via the AKS primality test). The algorithm certainly isn't rocket science, making it quite surprising that this wasn't found way earlier. The point is that it was common belief that $\text{PRIMES}$ is not in $\text{P}$, so there was not a big search for such an algorithm.
Another result that was discovered late is the construction of the Heptadecagon (regular seventeen-sided polygon) by compass and unmarked straightedge, found by Gauß in 1796.
Wikipedia describes it as "...the first progress in regular polygon construction in over 2000 years"
Remarkable enough, considering that is a purely geometrical result, and the Greek already had everything that was necessary for achieving it in their toolbox.
This is not a theorem but a result as you require that was discovered "late".
As Erdős mentions:
"At that time, mathematicians of the seventeenth and eighteenth centuries found pairs of odd amicable numbers such as $(12285,14595)$.
Curiosly, however,the smallest pair after the one known from antiquity $(1184,1210)$ was found only in 1866 by a 16-year old student Niccolo Paganini".
Theorem $\big($Sylvester-Gallai$\big):$ For any $n$ points in $\mathbb{R}^2$, not all collinear, there exists a line passing through exactly two of them.
$\rm{Proof}:$ Pick any $2$ points and draw a line $\ell$ through them: suppose a $3^{\rm{rd}}$ point lies on the line $($else we are done$)$ and pick the closest point $p\not\in\ell$ to the line, at a distance $\delta$ say. Of our $3$ points $\in\ell$ a pair lies on one side of $p:$ draw a line $\ell'$ through $p$ and the furthest of the pair from $p$. The distance $\delta'$ between $\ell'$ and the second point is $<\delta$.
$\qquad\qquad\qquad\quad$
The statement was proposed first by Sylvester in $1893$ and independently by Erdős in $1943\;($rather surprisingly, Erdős could not find a proof$)$. It was proven the following year by Gallai.
$\big[\mathbf{Note:}$ the statement can fail in other fields: for instance, in $\mathbb{P}^2(\mathbb{C})$ see the Hesse configuration$\big]$
Incredible proof.
– KalEl Mar 30 '14 at 05:20A Dandelin sphere of a conic section will touch the plane of the conic section at a focus of the conic section. This pretty and useful fact was discovered only in 1822, but there is no reason why it could not have been in the treaties of Apollonius of Perga.
One possible answer would be the discovery of Strassen algorithm in 1969. This is a way to multiply large matrices together in (slightly) faster than standard time.
Although it may not be the fastest algorithm for matrix multiplication the majority of faster variants stem from this one idea.
It may not be a simple proof but it is very late.
I'd say the fact that zero took until sometime after the 600s before it was used in math takes the cake here.
Grothendieck said that he found it embarassing that it took humanity so long to define a group.
It took 2000 years for anyone to recognize that Euclid's Parallel Postulate didn't follow from his other four axioms. It has always been surprising to me that people tried for so long to prove the parallel postulate followed from the other axioms and no one seemed to consider the possibility that it was independent of them. The ultimate realization that there exist models of geometry in which the converse holds has blossomed into an enormous area of mathematics rife with deep and interesting results.
Pascal's theorem was discovered in 1639 or 1640, but its projective dual, Brianchon's theorem, was only discovered in 1806. So it took about 167 years despite the fact that Pascal already realized the projective nature of his theorem.
Generally the JSJ-decomposition of 3-manifolds is considered to be discovered "late" (by $40$ years, but that is a long time in modern maths!).
From Allen Hatcher's Notes on Basic 3-Manifold Topology,
Beyond the prime decomposition, there is a further canonical decomposition of irreducible compact orientable 3 manifolds, splitting along tori rather than spheres [the JSJ-decomposition]. This was discovered only in the mid 1970’s, by Johannson and Jaco-Shalen, though in the simplified geometric version...it could well have been proved in the 1930’s. (A 1967 paper of Waldhausen comes very close to this geometric version.) Perhaps the explanation for this late discovery lies in the subtlety of the uniqueness statement. There are counterexamples to a naive uniqueness statement, involving a class of manifolds studied extensively by Seifert in the 1930’s. The crucial observation, not made until the 1970’s, was that these Seifert manifolds give rise to the only counterexamples.
For context, JSJ-decompositions are the starting point for Perelman's proof of Geometrisation, which implies the Poincare conjecture. His subsequent shunning of a fields medal and $1m prize is oft-told...
The first thing that came to my mind was the Kepler conjecture. While it's not a straightforward or simple proof, the fact that fruitsellers stacked their apples optimally for thousands of years until it was proven that they did it is somewhat surprising.
The Mason-Stothers theorem (aka the $abc$ conjecture for polynomials): if $f(t)$, $g(t)$, and $h(t)$ are nonzero polynomials over a field satisfying $f(t) + g(t) = h(t)$ with $f(t)$ and $g(t)$ being relatively prime and the three polynomials are not all constant, then $$ \max(\deg f, \deg g, \deg h) \leq \deg({\rm rad}(fgh)) - 1, $$ where ${\rm rad}(F)$ for a nonzero polynomial $F$ is the product of its irreducible factors (set it to be $1$ if $F$ is constant). Strictly speaking, the version I wrote is for a field of characteristic $0$. In characteristic $p$, the condition that at least one of the polynomials is nonconstant should be replaced with at least one of them having nonzero derivative.
This theorem was proved only in the 1980s, independently, by Mason and Stothers. Its proof uses just some elementary calculations with derivatives of polynomials. There was a proof given by Silverman that shows this result is, to use the OP's term, a "straightforward corollary" of the Riemann-Hurwitz formula, so in principle the theorem could have been formulated in the 19th century, but the simple statement of this theorem did not appear until Mason and Stothers as far as I'm aware.
On somewhat the same theme, another example of the type being requested is Belyi's theorem. It says that for a smooth projective algebraic curve $C$ over the complex numbers, if $C$ is defined over the algebraic numbers then $C$ admits a covering of the Riemann sphere ramified over at most three points. (The converse was known earlier.) I say this is somewhat the same theme because Belyi's theorem is closely related to when the inequality of the Mason-Stothers theorem is an equality. Like the Mason-Stothers theorem, the proof of Belyi's theorem is surprisingly low-tech compared to what anyone would imagine when hearing the statement of the theorem.
I got three of them for you.
Trisecting the angle, doubling the cube, and squaring the circle. They were three famous geometric problems from ancient Greece. They were concerned about whether or not these could be done with compass and straightedge. They all ended up being impossible, but it wasn't proven for 2000 years.
Trisecting the angle was disproved in 1837 using Galois theory
Doubling the cube was proved impossible in 1839 by showing that $\sqrt[3] 2$ is not a constructable number.
Squaring the circle was proved impossible in 1882 by showing that $\pi$ is transcendental
Find an explicit bijection between $\mathbb N$ and $\mathbb Q^{+}$.
The countability of $\mathbb Q^{+}$ had been known for a century, but an explicit bijection was not known until, in $1989$, Yoram Sagher noticed a rather simple explicit bijection between $\mathbb N$ and $\mathbb Q^{+}$.
Let $\frac{m}{n} \in \mathbb Q^{+}$ with $gcd(n,m)=1$, and let $q_1, q_2, \dots, q_n$ be the prime factors of $n$. Then $$f\left(\frac{m}{n}\right) = m^2n^2/(q_1q_2\cdots q_n)$$ is the desired bijection. Given the simplicity of $f$, it is surprising that no one had noticed it before.
Note that the inverse is (easily) computable, so you say what the $n$-th rational number listed is for any $n$.
The Perko pair:
Since the first knot tables published around 1900, these two were thought to be different knots. The mistake slipped by Alexander's (1920s), Conway's (1960s), and Rolfsen's (1970s) tabulating efforts. It was only in 1974 that Kenneth Perko noticed that these two diagrams belong to one and the same knot.
(for cuteness sake, the picture above is from Perko himself)
Brouwer's demonstrations of the difficulties with classical logic and the excluded-middle principle (tertium non datur), which had been used in mathematics and logic for more than 2000 years.
The lateness of discovery was due to the increasing frequency and complexity of infinitary constructions in 19th-century analysis (and the relatively more concrete arguments before that time), and the concomitant pressure to develop precise and consistent ways of handling them. The extension of principles like Excluded Middle from finite to infinite situations is where the problems isolated by Brouwer most clearly appear.
Smale's paradox: in topology, the fact a sphere in R3 can be turned completely inside out without tears or sharp creases (self-intersection is allowed) was only indirectly proved in 1957 and an actual example of the process wasn't found until 1961.
See http://en.wikipedia.org/wiki/Smale%27s_paradox and http://www.geom.uiuc.edu/docs/outreach/oi/history.html
Various irrationality and transcendence results have already been posted, but it is interesting to see that the mere existence of transcendental numbers was not proven until the nineteenth century. Of course the existence of irrational numbers was already known in ancient Greece, but it took until 1844 before we first knew with certainty that there exist transcendental numbers.
The notion of algebraic and transcendental numbers was not yet available in ancient Greece. It seems that the first mention of the term transcendental was made by Leibniz in the 17th century, although he was more interested in transcendental functions rather than transcendental numbers. As Bourbaki puts it (Elements of the History of Mathematics, page 74):
“The definition that Leibniz gives of "transcendental quantities" [...] seems to apply more to functions than to numbers (in modern language, what he does reduces to defining transcendental elements over the field obtained by adjoining to the field of rational numbers the given numbers from the problem); it is however likely that he had a fairly clear notion of transcendental numbers (even though these latter do not appear to have been defined in a precise way before the end of the XVIIIth century); [...]”
The first proof of the existence of transcendental numbers was given by Liouville in 1844, who constructed a class of numbers which he then proved to be transcendental (now known as the Liouville numbers). The Louiville numbers are a cornerstone in the field of Diophantine approximation, which has since grown into a rich and active field of study in contemporary mathematics.
Nowadays of course we get the existence of transcendental numbers as an easy corollary of a famous theorem of Cantor's, which states that $\mathbb{R}$ is uncountable. Since the set of algebraic numbers is only countable, it follows that there must exist (uncountably many) transcendental numbers. Knowing this simple proof, I was very much surprised to learn that the existence of transcendental numbers had only been settled for 30 years when Cantor first proved $\mathbb{R}$ to be uncountable.
(Interestingly, Cantor's first proof of the uncountability of the real numbers predates his famous diagonal argument by some 17 years. See also Robert Gray, Georg Cantor and Transcendental Numbers, The American Mathematical Monthly, Vol. 101, No. 9 (Nov., 1994), pp. 819-832, at the time of writing also available in its entirety at the website of the Mathematical Association of America.)
I'm not sure this is a real answer, but Wikipedia claims that the Möbius strip was invented in 1858, by Möbius. I find it incredible that there is any property of the strip that was still undiscovered by 1858, so to whatever fraction of Wikipedia's claim is true I think qualifies as a surprisingly late discovery.
The Meyers Serrin theorem that $ H=W $ (see their paper with exactly this equality as title) came only decades after people had proven all kinds of things with both $ H $ and $ W $, and is so elementary that you can ask second year students to prove it as exercise (extra credit, to be fair)
How about the Jordan curve theorem? It seems obvious at first sight, but it is quite complicated to prove.
Another result that seems to have appeared late is the AKS primality test ( http://en.wikipedia.org/wiki/AKS_primality_test ). Although the field of computational compexity is relatively new, one could have expected that a proof for the existance of a deterministic polynomial-time primality-proving algorithm should have appeared earlier - given the fact that some of the most intelligent people in the world have studied prime numbers intensively for thousands of years.
The Futurama Theorem: Regardless of how many mind switches between two bodies have been made, they can still all be restored to their original bodies using only two extra people, provided these two people have not had any mind switches prior (assuming two people cannot switch minds back with each other after their original switch).
The one that bumps up in my mind is Erdős-Mordell inequality. It states that for a point $ p $ in a triangle $ ABC $, with $\alpha ,\beta,\gamma $ denoting the distance of $ P $ to the 3 sides,respectively, then $ AP+BP+CP\geq 2 (\alpha + \beta +\gamma)$ This wasn't thought of until Erdős conjectured it in 1935.
P.s.People think Erdős might have been inspired by Euler's theorem for triangles, which itself is a 'shame' of Euclid.
Answer
Mirsky's theorem (1971) in order theory has been discovered late.
Background
In 1950, Robert Dilworth published what is today known as Dilworth's theorem:
In any finite poset $(X,\leq)$, the size of the largest antichain equals the minimum number of blocks of a partition of $X$ into chains.
By now, several proofs are known, but for the "hard" implication "$\Rightarrow$", all of them are somewhat involved inductive proofs.
There is a pretty natural "dual" of the statement, today known as Mirsky's theorem:
In any finite poset $(X,\leq)$, the size of the largest chain equals the minimum number of blocks of a partition of $X$ into antichains.
Surprisingly, apparently no one had thought about that dual statement for about two decades, until it was discovered and published in 1971 by Leon Mirsky. All the more as the proof turned out to be much simpler than everything known for Dilworth's theorem. For the hard implication "$\Rightarrow$", there is a direct construction of the partition as the preimages of the map $$X \mapsto \mathbb{N},\quad x\mapsto \text{size of the largest chain with maximal element }x.$$ Nothing comparable is known for Dilworth's theorem.
More details on Mirsky's 1971 publication
Mirsky's theorem was published in the article [L. Mirsky: A dual of Dilworth's decomposition theorem, The American Mathematical Monthly 78[8] (1971) 876-877.]
From the article:
We owe to Dilworth [1] the following well-known and important decomposition theorem:
Theorem 1 [...]
It may be of some interest to note that this statement remains valid if the roles of chains and antichains are interchanged. Thus we have the following result:
Theorem 2 [...]
Thus, in a formal sense, Theorem 2 may be regarded as a 'dual' of Theorem 1. However, as we shall see, the proof of the dual result is considerably easier than that of Dilworth's original theorem. [...]
The Deduction Meta-Theorem for slews of logical calculi, and slews of conditional proofs in ordinary mathematics which can get said to have an interaction with it.
James A. Garfield (20th President of the United States) found 1875 (~2175 years after Euclid) a quite simple, yet clever trapezoid proof of the Pythagorean theorem. Here is a video explaining it:
An example of this that jumps to my mind is Fermats last theorem, for which was conjectured by him in 1637 but only proved in 1995 despite many years work by countless mathematicians. But this proof (obviously) is not simple nor straightforward. I don't know how important that criteria is on our answers, but it is an example of a result that many believed to be true for a very long time, yet to be unable to be proved. Fermats last theorem is that,
For a,b and c being positive intergers
$a^n+b^n\neq c^n$ For n bigger than 2
