The envelope that reached G.H. Hardy in Cambridge did not look like the beginning of one of mathematics’ most famous collaborations. It looked like unsolicited mail from a stranger making impossible claims.

Its sender introduced himself as a poorly paid clerk in the accounts department of the Madras Port Trust. He had no university degree, no post at a research institution and no obvious route into the small professional world to which Hardy belonged. What he did have was page after page of formulas.

The letter was dated 16 January 1913. Its author, Srinivasa Ramanujan, stated results involving infinite series, integrals and continued fractions with almost none of the demonstrations that would normally let a mathematician decide whether they were true.

Hardy was suspicious. He had received fanciful mathematical claims before, and an unknown correspondent could be mistaken, dishonest or simply impossible to understand. Yet some of these statements were recognisable, while others were unlike anything he had seen.

The story is often told as a single evening’s reversal. Hardy took the pages to his younger colleague J.E. Littlewood; the two worked through them after dinner; by midnight, they knew the clerk in Madras was a mathematician of exceptional originality.

That scene is substantially supported by the later accounts, though not preserved as a timestamped diary. What Hardy himself recorded is more valuable than the clock time: several formulas defeated him, but they were too coherent and imaginative to dismiss. Suspicion had forced him to look closely, and looking closely changed both men’s lives.

Unknown in Cambridge did not mean undiscovered in India

Ramanujan was born in 1887 in Erode, in what was then the Madras Presidency, and grew up mainly in Kumbakonam. As a teenager he became absorbed in mathematics, especially after encountering G.S. Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics.

Carr’s book compressed thousands of mathematical statements into a form intended for students preparing for Cambridge examinations. Many results appeared with little or no proof. For a young man studying largely on his own and with limited access to advanced books, it offered an immense territory to explore. It also helped shape Ramanujan’s habit of recording conclusions more fully than the route to them.

His devotion came at a cost. Ramanujan won a college scholarship, then lost it after neglecting subjects other than mathematics. He failed later examinations, left without a degree and spent years trying to secure enough income to continue his work.

It is tempting to describe him as wholly isolated until Hardy rescued him, but that erases the people who recognised his ability in India. V. Ramaswamy Aiyer, R. Ramachandra Rao, S. Narayana Iyer, E.W. Middlemast and Madras Port Trust chairman Francis Spring all encouraged him or helped him find support.

By 1911, Ramanujan had published a long paper on Bernoulli numbers in the Journal of the Indian Mathematical Society. On 1 March 1912, he began work as a Class III, Grade IV accounting clerk at the Port Trust. Colleagues allowed him room to pursue mathematics when his assigned work was finished.

So the author of the 1913 letter was unknown to Hardy, not unknown to everyone. He was already part of a small Madras network struggling to connect unusual work with an institution able to evaluate it.

Two unanswered approaches came before Hardy

Ramanujan did not choose Hardy out of nowhere. He had seen Hardy’s 1910 book Orders of Infinity, which dealt with questions close to his own interests. Before writing to him, Ramanujan had approached two other Cambridge mathematicians, H.F. Baker and E.W. Hobson. Neither produced the opening he needed.

The third letter began with a plain introduction. Ramanujan described his job, lack of conventional university education and independent investigation of divergent series. Then the mathematics took over.

The first packet is generally described as nine pages of formulas accompanying the covering letter. Hardy later said Ramanujan’s first letters contained the bare statements of about 120 theorems, mostly formal identities drawn from his notebooks. The important distinction is that 120 refers to the early correspondence as a whole, not necessarily to the single envelope dated 16 January.

“Almost no proofs” is accurate as a description of the pages Hardy received. It is not evidence that Ramanujan reached every conclusion by mystical intuition or could prove none of them. The surviving notebooks mostly preserve finished results, not every calculation performed along the way.

Why a genuine genius could resemble a crank

To Hardy, the presentation created a difficult diagnostic problem. Mathematical authority does not come from confidence or quantity. A page can contain dozens of impressive-looking equations and still say nothing true.

The collection was also uneven. Some results were already known in European mathematics, although Ramanujan may have rediscovered them independently. Some could be checked with existing techniques. A few were false or misleading. Others were new, deep and extremely difficult.

In his 1940 lectures on Ramanujan, Hardy reconstructed how a professional mathematician might react to such a letter. He recognised one formula as classical and another as having appeared in earlier work. He managed to prove two integral formulas, but with more difficulty than expected.

Then he reached results on continued fractions that “defeated me completely.” Hardy concluded that they could only have come from a mathematician of the highest class. Their internal quality mattered: if they were fabricated, the fabricator would need almost as much mathematical imagination as the genius supposedly being faked.

This was not blind faith. Hardy’s later analysis of 15 representative claims found a complicated picture. Thirteen were correct and significant, a fourteenth was false but fruitful, and a fifteenth was technically true yet misleading. The letter earned attention because an expert could discriminate within it, not because every line was accepted as revelation.

What happened before midnight

The best-known biographical version has Hardy set the letter aside, then return to it after dinner. He and Littlewood spent roughly two or three hours examining the formulas. Before midnight, the two were convinced that the unknown sender possessed extraordinary ability.

We should be modest about the scene’s cinematic precision. There are no meeting minutes recording the instant suspicion became belief. Hardy’s own later mathematical account confirms that he considered every possibility, tested what he could recognise and arrived at essentially the same judgment. The evening narrative is a reconstruction of that process, not a transcript.

It is also easy to make their recognition sound instantaneous. They did not prove the deepest results that night. Some would require years of work by Ramanujan, Hardy, Littlewood and later mathematicians. What they established was that the pages demanded a serious reply.

On 8 February 1913, Hardy wrote back. He said he was exceedingly interested, but insisted that he needed proofs before he could judge properly. That request was not a failure to appreciate Ramanujan. In mathematics, proof is what turns a striking assertion into knowledge other people can inspect and use.

ScienceBlog’s earlier report on high-school students who found new proofs of an ancient theorem rests on the same distinction. A result may be familiar while a proof is new; a result may look plausible while no proof exists; and a proof can reveal why something is true in a way the answer alone cannot.

The letter opened a door, but many people held it open

Hardy’s response gave Ramanujan something he had not received from the first Cambridge mathematicians he contacted: informed attention. He also began contacting officials who might help.

In May 1913, the University of Madras awarded Ramanujan a research scholarship. Bringing him to England took longer. Travel across the sea conflicted with religious and family expectations, and Ramanujan initially declined. Cambridge mathematician E.H. Neville later met him in Madras and helped make the journey possible.

Ramanujan sailed in March 1914 and reached England in April. At Cambridge, he began the difficult exchange that the letter had foreshadowed. Hardy and Littlewood could offer the language and discipline of modern proof. Ramanujan brought results and methods that repeatedly led them into unfamiliar territory.

The partnership should not be reduced to a simple story of Cambridge teaching an untrained outsider how mathematics worked. Hardy was trying to supply missing parts of Ramanujan’s education without flattening the originality that made his work valuable. Littlewood later recalled how difficult systematic teaching became because almost every topic prompted a rush of new ideas.

Nor was this solely a British discovery of Indian talent. Indian mathematicians, patrons and colleagues had already recognised Ramanujan and kept him afloat. Hardy’s great contribution at the start was to recognise what was in front of him and use his institutional position to make a larger mathematical life possible.

That pattern has appeared elsewhere in scientific history. ScienceBlog’s account of Katherine Johnson hand-checking the orbital calculations for John Glenn likewise shows that ability and recognition are not the same thing. Institutions decide whose mathematics reaches the centre and whose remains at the edge.

The almost proofless pages kept generating proofs

Ramanujan spent nearly five years at Cambridge. With Hardy, he developed a powerful asymptotic formula for the partition function and helped create what became known as the circle method. His work ranged across continued fractions, infinite series, modular equations and number theory.

He was elected a Fellow of the Royal Society in 1918 and became the first Indian Fellow of Trinity College later that year. Poor health forced him to return to India in 1919. He died on 26 April 1920, aged 32.

His mathematics did not stop moving. Researchers spent decades proving, correcting and extending results from his notebooks. His final work on mock theta functions, described in a letter to Hardy shortly before his death, became connected to areas of mathematics that did not yet exist in his lifetime.

A MacTutor history of Ramanujan’s life and work captures both sides of the record: the depth of his originality and the gaps created by limited formal education. Romantic accounts often keep only the first side, as if proof would make the story less wondrous. In fact, the long labour of proof is part of what shows how rich the original claims were.

The letter of 16 January 1913 did not manufacture a genius, and one Cambridge evening did not verify his mathematics. The letter carried years of work across an institutional divide. Hardy and Littlewood’s achievement was smaller but still rare: they looked past an implausible presentation, tolerated uncertainty and recognised that the stranger deserved to be taken seriously.

Before midnight, they did not know where every formula came from or whether every one was right. They knew something more immediately useful. The pages could not be thrown away.