Ole Romer gave us one of the first accurate evaluations for the speed of light.

When Sir Isaac Newton produced his revolutionary work on calculus, the mathematics of integration and differentiation, he was also able to give us a far better understanding of the force of gravity and present equations that very accurately described the motion of projectiles on Earth, the motion or ‘orbits’ of the planets around the sun and the orbit of the moon around the Earth and so on.

However in Newton’s era the transmission of light from one place to another was generally believed to be instantaneous and similarly the force of gravity was also assumed to act instantaneously.

It was only in 1676 that the Danish astronomer Ole Rømer's careful observations of the moons of Jupiter that revealed that light did not travel infinitely quickly, but that it must in fact have an actual, and finite speed.

Romer was looking closely at the orbits of Jupiter's moons to see if they could be used as a kind of universal clock, that would be the same for all observers on Earth. In itself a very clever idea.

At  http://physics.info/light/  it is explained...

Measurements of the speed of light.

Galileo Galilei (1564-1642) Italy (Tuscany): In fact I have tried the experiment only at a short distance, less than a mile, from which I have not been able to ascertain with certainty whether the appearance of the opposite light was instantaneous or not; but if not instantaneous it is extraordinarily rapid.

Ole Christensen Rømer (1644-1710) Denmark. "Démonstration touchant le mouvement de la lumière trouvé par M. Roemer de l'Académie des Sciences." Journal des Scavans. 7 December 1676. Rømer's idea was to use the transits of Jupiter's moon Io to determine the time. Not local time, which was already possible, but a "universal" time that would be the same for all observers on the earth, Knowing the standard time would allow one to determine one's longitude on the earth — a handy thing to know when navigating the featureless oceans.

Unfortunately, Io did not turn out to be a good clock. Rømer observed that times between eclipses got shorter as earth approached Jupiter, and longer as earth moved farther away. He hypothesized that this variation was due to the time it took for light to travel the lesser or greater distance, and estimated that the time for light to travel the diameter of the Earth's orbit, a distance of two astronomical units, was 22 minutes.

Wikipedia explains,

http://en.wikipedia.org/wiki/Ole_R%C3%B8mer#R.C3.B8mer_and_the_speed_of_light

Rømer and the speed of light

The determination of longitude is a significant practical problem in cartography and navigation. Philip III of Spain offered a prize for a method to determine the longitude of a ship out of sight of land, and Galileo proposed a method of establishing the time of day, and thus longitude, based on the times of the eclipses of the moons of Jupiter, in essence using the Jovian system as a cosmic clock; this method was not significantly improved until accurate mechanical clocks were developed in the eighteenth century. Galileo proposed this method to the Spanish crown (1616–1617) but it proved to be impractical, because of the inaccuracies of Galileo's timetables and the difficulty of observing the eclipses on a ship. However, with refinements the method could be made to work on land.

After studies in Copenhagen, Rømer joined the observatory of Uraniborg on the island of Hven, near Copenhagen, in 1671. Over a period of several months, Jean Picard and Rømer observed about 140 eclipses of Jupiter's moon Io, while in Paris Giovanni Domenico Cassini observed the same eclipses. By comparing the times of the eclipses, the difference in longitude of Paris to Uranienborg was calculated.

Cassini had observed the moons of Jupiter between 1666 and 1668, and discovered discrepancies in his measurements that, at first, he attributed to light having a finite speed. In 1672 Rømer went to Paris and continued observing the satellites of Jupiter as Cassini's assistant. Rømer added his own observations to Cassini's and observed that times between eclipses (particularly those of Io) got shorter as Earth approached Jupiter, and longer as Earth moved farther away. Cassini made an announcement to the Academy of Sciences on 22 August 1676:

According to wikipedia, the announcement made by Romer was...

This second inequality appears to be due to light taking some time to reach us from the satellite; light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of the terrestrial orbit.[3]

(My Underlining)

My problem here is that the speed of light is a very critical component of physics, and it is used throughout Relativity and cosmology etc. And here in one of its first scientific descriptions it is said to 'Take Time' to travel (from Io to Earth).

But, as far as I am aware, Romer (or anyone else) did not prove the existence of a thing called Time, or of the past or of the future. Nor did he prove that as things move, e.g. Light or anything else, they 'Take' Time to move, or that Time exists and happens to 'pass' as things move.

Diagram: Io orbits Jupiter, images of this travel to Earth at a finite speed, as the Earth steadily rotates on its axis. We can compare these details, and express and 'speed' in terms of any other - But, despite what seems to be habitually assumed by experts and laypeople alike, this does not prove there is a 'past', a 'future', or that 'Time' exists, and passes as things move.

When Romer says...

light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of the terrestrial orbit.

In my opinion this 'ten to eleven minutes' just describes how, as light is travelling through space at its own special speed, the Earth is also rotating, and in this case, as light travels 'a distance equal to the half-diameter of the terrestrial orbit' the Earth rotates through 2.5 to 2.75 degrees.

This figure reached thus, 360 degrees, divided by 24 'hours', give 1 'hour' = 15 degrees.

15 degrees / 60 'minutes' = .25 degrees

0.25 * 10 or 0.25 * 11 = 2.5 or 2.75 degrees.

And this is not the same thing expressed differently, because stating light 'takes' 10 or 11 'minutes' to travel suggests that there is an invisible fourth dimension called time, comprising of an invisible past and future etc.

While 'degrees' just suggests that things are as we directly observe. Light moves, the Earth spins, and we can choose to very usefully, compare these things if we wish.

Astronomers and physicists realising that light had a speed were then led to consider and explore whether ‘gravity’ might also have a finite speed, and what it might mean if gravity does have a speed, particularly when we consider that while ‘light’ is something that ‘goes from A to B’ say when you switch on a torch, gravity is very different, because gravity is  ‘already in place’ all over the universe, so what could it mean for ‘gravity to have a speed?.

Rømer’s discoveries began as he was observing the motion of the moons of Jupiter through the newly created telescopes of his day. What Rømer realised is that while it made sense that the moons of Jupiter should be orbiting the planet at a regular rate this did not appear to be the actual case, and in fact the moons seemed to ‘lag’ behind where they might be expected to be when Jupiter was far from earth and appeared to be ahead of where they were expected to be when Earth and Jupiter were closer together.

Rømer realised that sense could be made of these observations and estimates or predictions about where the moons of Jupiter should appear to be in their orbits could be made very accurate, if any calculations made were adjusted to take into consideration how near or far from the Earth Jupiter actually was when it was being observed, and the ground breaking idea that light itself had a speed.

Rømer probably began his reasoning by making the safe assumption that the moons of Jupiter orbit the planet (or gas giant) at essentially a steady, constant rate. We can see why this is a sensible assumption because If you swing a small weight around your head on a piece of string, or better still set a ‘pendulum’ swinging not just side to side but in a circular motion you can easily observe how this circular motion should be very similar in many ways to the circular motion of a moon orbiting a planet. And you can extend the analogy to see that similarly unless there is some particular reason why it should be otherwise the speed of a ‘circling’ mass will remain constant.

From this observation, and by observing the way our own moon orbits the earth in a very regular way we can see that it is safe to assume that a large body, or moon orbiting another large body or planet in open space will, as Newton’s equations described, continue to do so. And do so at an unchanging rate unless something very significant causes it to do otherwise.

'Taking time' vs 'travelling at a speed'.

If we say something 'took time' to travel from A to B, then we are implying that a thing called 'Time' exists in some way. If we just say something 'travelled at a finite speed', then we are saying nothing more that 'that things can exist and move'. This is not a matter of semantics, click the above link for the distinction.

While it can be assumed that any particular moon orbiting Jupiter will actually be doing so at a very regular rate Rømer realised that the images of Jupiter that we see did not arrive on earth instantaneously but assumed they ‘took time’ to reach Earth.

Romer then made calculations relating the distance from the earth to jupiter, and whether the moons he expect to see appeared sooner or later than expected. Mathematically he could then express the speed of light. He would have done so ultimately in terms such as 'miles per hour', or miles per second. And so seemed to show that 'light took time' -so many seconds - to travel from jupiter to earth.

However, really, all he showed was that light did not travel to earth instantaneously, but, while light was travelling from jupiter to earth, the light moved, and the earth rotated. The distance from earth to Jupiter varies greatly as they both orbit the sun- which is why Jupiter and its moons are so useful for this calculation. But if Romer had calculated at one point that the light took say '30 minutes' to reach earth then really all he is saying is that at the light makes the journey - the earth rotates 7.5  degrees on its axis.

Another way of expressing the results would have been to say 'Light seems to travel around 643 thousand times as fast as a point on the earth's equator. This is expressing the same fact without using the assumption that time exists, or claiming to have proven that time exists.

Timeless light speed.

So, in a timeless view we would reword this explanation to say that the images from Jupiter ‘had to travel at a finite speed over a distance for us to see them’, and thus the images that we see ‘here’ on Earth are always or constantly misaligned with what is actually happening ‘there’ at Jupiter.

This difference in descriptions is not purely semantic because the first expression ‘the images take time to get to earth’ suggests the existence of a thing called time, with its past, present and future ‘baggage’ and numerous other ‘features’. While the second explanation suggests absolutely no more than the idea that images are formed, as the sunlight reflects off Jupiter and its moons, and that these images, in reality highly re-organised streams of photons, can travel at a speed, in all directions. Some of them travelling in the direction of Earth, and that’s it.

The images either exist or don’t exist, and are either travelling in various directions or not, but there is no suggestion that ‘time’ is needed or used.

Confusion can arise here if you insist that things cannot just move, but have to move ‘over’ a thing called time, and also insist on describing speeds in terms of ‘distance over time’ instead of describing speeds in terms of one thing moving faster or slower than another.

In the first case you might insist that the images from Jupiter travel to the earth at ‘300,000 km per second’, and therefore that ‘seconds exist’ and the images take time to get here and are ‘out of date’ when they arrive, and in the second case you would say, a ‘second’ is just another way of expressing the earth rotating through 0.000416 of a degree, or that the images from Jupiter travel ‘about 650,000 times faster than a point on the rotating earths equator’, and there is a significant distance for them to cover so we can see them.

In this second view the fact that the images of Jupiter’s moons we receive won't correlate with where the moons are is no more mysterious than the observation that if you are playing tennis your opponents racket carries on moving forward while the ball they have just struck moves across the court towards you.

If you and your opponent are very close together near the net their racket will be close to the position it was when it struck the ball you receive, if you are both far apart each right at your respective ends of the court, your opponents racket, and even the entire player will move a lot while the ball moves across the court to reach your racket. Both things, your opponent and the ball they strike, are just moving and there is no deeper mystery of time, and no ‘time delay’ is ever seen.

However you look at the situation Rømer realised that as we look at and see an image of Jupiter how the planet and its moons actually ‘are’ is a different thing to how the images of Jupiter that are arriving at earth actually ‘are’.

Rømer therefore also realised that If we were located very close to Jupiter we would be seeing images of the planet and its moons that were very close to how the situation at Jupiter actually was, each moon might for example be only a few fractions of a degree further around its orbit than its image suggest, but if we were located millions of kilometres away the moons of Jupiter are able to continue much further in their orbits as ‘out of date’ images make their way into space.

Because Earth and Jupiter orbit the sun at different rates the distance between the two planets is constantly changing in a very orderly way, so by observing the difference between where he knew the moons of Jupiter must be, and where the images reaching earth showed them to be Rømer could see for example that as light travelled from Jupiter to Earth when their orbits happened to bring the two planets close together a particular moon of Jupiter might be able to complete one quarter of a complete orbit.

But when the planets were a greater distance apart the same moon was able to complete half an orbit. From comparing observations like this he was able to come up with a surprisingly accurate estimate for the speed of light, close to the speed we describe as 300,000km per ‘second’ today. (a ‘second’ really just being a unit referring to a particular fraction of complete rotation of the Earth on its axis).

Einstein deduced that because there do not seem to be any instantaneous reactions observed in the universe ‘gravity’ itself, or rather ‘ripples’ through fields of gravity, should also have a speed, and therefore a finite speed. And astronomical observations show that this speed turns out to be identical to the speed of light, more accurately the speed given by the number known as ‘Einstein’s constant’ and referred to by the symbol ‘C’.