In the early nineteenth century, two lingering difficulties in Euclid's Elements were addressed and resolved. The first was the troubling role of the parallel postulate. Three men, from Germany, Hungary, and Russia, worked in parallel, effectively decomposing and then recomposing geometry. They showed that the postulate is just one of several valid assumptions that can be used in deductive geometry. This at first caused widespread consternation, which metamorphosed over time into acclamation.

The second difficulty was to prove the converse of Euclid's assertion that two polygons have equal area if it is possible to decompose one into pieces that recompose to form the other. Three different men, from Germany, Hungary, and England, worked in parallel to decompose and then recompose polygons. But their result caused no consternation and produced at best fleeting fame. Too bad for dissections that the parallel was not stronger!