Mathematicians are human, just like the rest of us, which means that they sometimes make mistakes. As their work is read by other mathematicians, both during peer review and after publication, we might expect errors to be quickly spotted and addressed. But a fatal mistake in Alfred Kempe's 1879 "proof" of the Four Color Theorem remained unnoticed for over a decade. The error was finally uncovered by Percy Heawood in 1890.
The Four Color Theorem states that for any map of contiguous countries drawn on a plane only four colors are needed to ensure that adjacent countries are given different colors. In the heart of his “proof,” Kempe had to consider a number of different cases of possible map configurations. To tackle these cases, he invented a new mathematical tool called Kempe chains. However Heawood discovered that Kempe's treatment of one of these cases didn’t work and presented an example to show how his reasoning failed.
Heawood's discovery did not mean that all was lost, though. First, Heawood showed that Kempe's work was enough to establish a weaker result called the Five Color Theorem which says that only five colors are needed to color a map in the required way. Second, although Kempe's proof was flawed, the Four Color Theorem was true. It was first proved successfully, though somewhat controversially, using a computer by Kenneth Appel and Wolfgang Haken nearly a century later in 1976. And Kempe’s ideas played a significant role in their work. So while Kempe made a mistake in his “proof,” it still contained valuable mathematics.