The image above doesn't look like much, but it's actually the remenants of a device known as the Antikythera Mechanism. Found in debris of a shipwreck, it ended up being one of the earliest known computers (for a given definition of "comptuer").
The Antikythera Mechanism was able to predict the motion the heavens as well as eclipses. The sophisticated device was made of bronze and used gears to make predictions.
For such a complicated device, you might expect that it was made around the time that many other such devices using gears (such as clocks) were being developed (around the 1600's). But this device ended up being much, much older. Estimates place the date of it's creation at 150-100 BCE!
Recently, the device has been examined using X-rays to make generate a 3-D image. The new study revealed a wealth of new incriptions.
Some great interactive images of the device can be found here. Also, for more information, and a really nifty Java applet showing the functioning of the Sun/moon gears try here.
One of the earliest. They probably used an abacus to figure out how to make it work.
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