Which astronomer discovered both Martian moons in 1877?

The astronomer who discovered both Martian moons in 1877 was Asaph Hall.

Working at the United States Naval Observatory, Hall discovered Deimos on August 12, 1877, and Phobos six days later on August 18. He was searching during a particularly favorable opposition of Mars, when the planet was relatively close to Earth. His wife, Angeline Stickney Hall, encouraged him to continue the difficult search when poor visibility made the task frustrating.

The moons were named after the companions of Ares, the Greek god of war: Phobos means fear or panic, while Deimos means dread or terror. Their tiny size and close orbits made them difficult to detect through telescopes, especially against the glare of Mars.

Hall is sometimes confused with Percival Lowell or Giovanni Schiaparelli, who became famous for studying Martian surface markings and canals. Christiaan Huygens observed important features on Mars much earlier, but he did not discover either moon. Hall also studied planetary rotation, satellite orbits, and stellar positions.