His only original contribution to geometry was the proof that any geometrical construction[?] which can be done with ruler and compasses can also be done with compasses alone. He published his proof in the book Euclides Danicus, Amsterdam, 1672.
Although the book was included in bibliographies of mathematics, nobody troubled to examine it, and it was totally overlooked for 250 years. The result was instead credited to the Italian Lorenzo Mascheroni, who independently delivered a proof a hundred years later (1797). Only in 1928, when a young student of mathematics found a copy in a secondhand bookshop in Copenhagen, did Mohr's achievement gain recognition. The book was reprinted in facsimile that year.
Mohr published his Euclides Danicus simultaneously in a Danish and a Dutch edition (each with a long subtitle in the respective language). One would have expected that a scientific work at the time would have been in Latin  in which case it would have been accessible to a wider circle of readers (for a similar case, see: Caspar Wessel). However, since he spent much time in the Netherlands, his choice of the national languages rather than Latin may have been inspired by the tradition started by the Dutch mathematician Simon Stevin (1548/91620).
Mohr was a friend of Tschirnhaus, and he spent his last few years as a guest in his house.
The Danish Mathematics competition is named in honour of Georg Mohr.
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