Here's how he did it:

* Observing Jupiter's moons: Rømer was observing the eclipses of Jupiter's moon Io. He noticed that the time between eclipses varied slightly depending on the Earth's position in its orbit around the Sun.

* The Earth-Sun distance: Rømer realized that the difference in timing was due to the changing distance between the Earth and Jupiter. When Earth was further from Jupiter, the light from Io's eclipse took longer to reach Earth.

* Calculating the speed: By meticulously measuring the time differences and knowing the Earth's orbital distance, Rømer was able to calculate the speed of light. His estimate was about 220,000 km/s, which is surprisingly close to the modern value of 299,792 km/s.

Significance of his discovery:

* Revolutionary: Rømer's work overturned the long-held belief that light traveled instantaneously. It was the first evidence that light had a finite speed.

* Foundation for further research: His work paved the way for further investigations into the nature of light, leading to the development of theories like Maxwell's electromagnetic theory of light.

* Impact on astronomy: Understanding the speed of light was crucial for calculating distances in space and accurately determining the positions of celestial objects.

In addition to his work on the speed of light, Rømer also:

* Made improvements to astronomical instruments: He designed and built several innovative astronomical instruments, including a meridian circle and a transit telescope.

* Contributed to the development of the concept of longitude: His work on the speed of light helped to advance the use of celestial observations for determining longitude.

Ole Rømer's work was a landmark achievement in physics and astronomy, forever changing our understanding of the universe.