A note at the end of a translation
In 1843 Ada LOVELACE was asked to translate an Italian article about Charles BABBAGE's unbuilt Analytical Engine. She delivered the translation with seven notes of her own appended, running to more than twice the length of the original.
The last of them, Note G, sets out step by step how the machine could calculate a sequence of numbers — generally considered the first algorithm written for a machine, a century before one existed to run it. Her larger point was the one Babbage had not made: that if the engine could manipulate numbers standing for quantities, it could manipulate symbols standing for anything at all, including music. That is the idea a computer actually runs on.
Who was allowed to do maths
HYPATIA taught mathematics and astronomy in Alexandria around 400 AD and was murdered by a mob in 415. Sophie GERMAIN, barred from the École Polytechnique because she was a woman, submitted work under a male student's name until Gauss discovered the deception and wrote to say her achievement was the greater for it. Emmy NOETHER lectured at Göttingen for four years under HILBERT's name, because the university would not appoint her.
Hilbert's response to the faculty has lasted: he pointed out that they were a university, not a bathhouse. The pattern behind these three names is not incidental to the history of maths — for most of it, the limiting factor on who made discoveries was who was permitted to sit in the room.
The clerk who wrote to Cambridge
In 1913 a shipping clerk in Madras with no formal training posted nine pages of theorems to G. H. Hardy at Cambridge. Most professors binned that kind of letter. Hardy looked at the results, decided they had to be true because nobody would have the imagination to invent them, and arranged passage to England.
Srinivasa RAMANUJAN produced thousands of results in the years that followed, many with no proof attached, simply asserted as things he knew. Mathematicians were still working through his notebooks seventy years after his death at 32, and finding them right. He said the formulae came to him from a family goddess in dreams, which nobody has ever managed to improve on as an explanation.
A margin too narrow
Around 1637 Pierre de FERMAT scribbled in the margin of a book that he had a truly marvellous proof of a certain claim about numbers, but that the margin was too small to write it in. Then he died, and the note was published by his son.
It took 358 years. Andrew Wiles finally proved it in 1994 using mathematics that did not exist in Fermat's lifetime, which strongly suggests Fermat's marvellous proof was wrong. The best guess is he spotted a flaw later and never went back to the margin. Generations of mathematicians lost years to that sentence.
The rest of the grid
The ancients first: PYTHAGORAS with his theorem, EUCLID whose textbook on shapes was still in use two thousand years later, ARCHIMEDES in his bath. Then the system-builders — GAUSS, EULER, LEIBNIZ, LAPLACE, RIEMANN, HILBERT and CANTOR, who proved some infinities are bigger than others and was widely told off for it.
And the ones with short, strange lives. Évariste GALOIS wrote out the foundations of a whole branch of algebra the night before dying in a duel at 20. George BOOLE invented the true-or-false logic every computer runs on, in 1854, for reasons that had nothing to do with machines. FIBONACCI brought Arabic numerals to Europe. TURING worked out what a computer was before anyone built one, and John NASH won a Nobel for work in game theory done at 21.