One of Galileo's most important discoveries he made through his telescope was the detection of the phases of the planet Venus. (Click here for one of Galileo's drawings from the Institute and Museum of the History of Science in Florence, Italy.) The primary importance of this discovery is that it proves that Venus does not orbit the Earth, as was believed in the Ptolemaic system, but it instead orbits the sun, as do all the planets in the Copernican system. In the Ptolemaic system, there is no way for Venus to go through a full set of phases. And yet it does. Game, set and match to Copernicus.
About 15 years ago, my first astronomy professor posed a somewhat-unfair question on an exam: How would the history of astronomy had been different if Venus were just a little closer to the Earth? (I say somewhat unfair, because grading on a hypothetical question is always highly subjective). The question arises out of the fact that the human eye can see details that are larger than about one minute of arc (1/60th of a degree), and when Venus is closest to Earth, it is just over one arcminute in size. Not only that, but when Venus is closest to Earth is also when it is going through its most extreme crescent phases.
So, if Venus were just a little closer to Earth, its crescent would be visible to the unaided eye, and even ancient astronomers would have known that it went through phases. Assuming nobody else would have figured it out, the ancient Greeks would have realized that phases meant Venus had to orbit the sun (they understood the geometry that causes the phases of the moon), and by assumption the rest of the planets would, too. Perhaps what we call the Copernican system might even have been proposed by Ptolemy, if only Ptolemy had known about the phases of Venus!
There are claims that some eagle-eyed ancient astronomers might have seen the phases of Venus, but it is difficult to prove this is the case. My gut feeling (which doesn't mean I'm right!) is that the Greeks would have seen the phases if it were possible, and there's no record that they did. It is very difficult to do a fair experiment these days, because we already know that Venus has phases, and even what phases the planet should have when we stare at it, and our brain is very good at telling us we see what we expect to see.
Over the next three weeks, the bright evening star that is Venus will grow to its largest angular size as its orbit carries Venus between the Earth and the sun. This is your chance to look for the phases of Venus. Even if you don't want to try to see it with your unaided eye, even small binoculars or a low-power telescope will easily show you the thin crescent that is Venus. Or, failing all else, check out this cool Astronomy Picture of the Day from last Friday, where you can see both the crescent moon and crescent Venus.