On September 27, 1822, Jean-Francois Champollion read a paper to the Academie des Inscriptions et Belles-Lettres in Paris, and afterwards wrote to the Academy's permanent secretary with the results. He had read the hieroglyphs. The letter, published as the Lettre a M. Dacier, set out an alphabet of phonetic signs and is dated to that month.
The problem had been open for about 1,400 years, and the reason it had not been solved was that everyone had assumed the wrong thing. Hieroglyphs were taken to be a writing of ideas, in which each picture stood for a thing or a concept, and on that assumption the number of signs and their arrangement made no sense. The Rosetta Stone, found in 1799 by soldiers in Napoleon's Egyptian expedition and handed to the British in 1801, had the same decree in hieroglyphic, demotic and Greek, which gave a way in but not the key.
He looked at the cartouches. On the Philae obelisk there was a cartouche in Greek and hieroglyphic, with Ptolemy in one and Cleopatra in the other, and he was able to match them sign for sign: P-T-O-L-M-Y-S and K-L-E-O-P-A-T-R-A. That gave him a set of phonetic values that worked for foreign names, which was the point at which it became obvious that the script was not purely ideographic.
He then took the step that mattered. Egyptian names, not only foreign ones, were written phonetically in cartouches, and the phonetic signs had been in use since the Old Kingdom, mixed in with the ideograms and the determinatives that everyone else had assumed were all there was. In 1822 he read the name of Ramesses, and Thutmose, and then the whole thing opened.
He was 31 and had been working on it since he was a boy in Figeac, where his brother had taught him and had got him a place at the College de France. He had learned Coptic, which is the last stage of the Egyptian language, from a priest, and Coptic is what let him know what the phonetic values should sound like.
The fight over credit was immediate and it was national. Thomas Young in England had worked out that the cartouches contained royal names and had got some phonetic values right before 1819, and Champollion was accused of using his work without acknowledgement. Young had also concluded that the script was mostly ideographic, which was the error.
Champollion was made a member of the Academy in 1830, given a chair at the College de France in 1831, and sent to Egypt in 1828. He died in 1832. Egyptian writing is now read from the earliest hieroglyphs of about 3200 BC to demotic in the Roman period, and the grammar and vocabulary are known well enough to read administrative papyri.