One of the better early attempts at dating the planet came from the ever-reliable Edmond Halley, who in 1715 suggested that if you divided the total amount of salt in the world's seas by the amount added each year, you would get the number of years that the oceans had been in existence, which would give you a rough idea of Earth's age. The logic was appealing, but unfortunately no one knew how much salt was in the sea or by how much it increased each year, which rendered the experiment impracticable.
The first attempt at measurement that could be called remotely scientific was made by the Frenchman Georges-Louis Leclerc, Comte de Buffon, in the 1770s. It had long been known that the Earth radiated appreciable amounts of heat—that was apparent to anyone who went down a coal mine—but there wasn't any way of estimating the rate of dissipation.
Buffon's experiment consisted of heating spheres until they glowed white hot and then estimating the rate of heat loss by touching them (presumably very lightly at first) as they cooled. From this he guessed the Earth's age to be somewhere between 75,000 and 168,000 years old. This was of course a wild underestimate, but a radical notion nonetheless, and Buffon found himself threatened with excommunication for expressing it. A practical man, he apologized at once for his thoughtless heresy, then cheerfully repeated the assertions throughout his subsequent writings.