Who discovered geometries where triangles add up to less than 180 degrees?

In hyperbolic geometry, the angles of a triangle always add up to strictly less than 180 degrees. Three people are credited with discovering such a geometry: Nikolai Lobachevsky, János Bolyai and, privately, Carl Friedrich Gauss. Their discovery came from denying Euclid's parallel postulate. In hyperbolic geometry, there are infinitely many lines through a point that never meet a given line. On a sphere you get the opposite: there are no parallel lines, because any two lines, which are great circles like the equator and the meridians, meet.

Ordinary triangles on a flat page total exactly 180 degrees. In a hyperbolic triangle the shortfall is called the defect, and the area of the triangle is its defect in radians times R², a constant. On a sphere it goes the other way: the right triangle that bounds an octant of the unit sphere has three right angles.

The discovery has a very human story. János Bolyai's father Farkas, who had spent years on parallels, wrote to him in 1820 not to try, calling it a bottomless night that extinguished all light and joy in his life. János carried on anyway, and in 1823 wrote to his father that he had created a strange new universe out of nothing. It was published in 1832 as an appendix to his father's book. Gauss's reply: to praise it would be to praise himself, because the content coincided with his own thoughts of thirty or thirty-five years. Gauss wrote in 1829 that he feared backlash if he published. Bolyai suspected his father had told Gauss, and later complained bitterly about Gauss's attitude.

So who gets the credit? Lobachevsky was first in print and the first to present his views to the world mathematical community. Bolyai worked independently. Gauss never published, and how far he preceded the other two is not quite clear.