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Cope's rule is one of several "biological rules" -- biological trends found in patterns across data about many plants and animals. Cope's rule states that population lineages tend to increase in size over evolutionary time. There are numerous examples throughout evolutionary history -- mammals increasing in size after the demise of the dinosaurs; reptiles increasing in size after the demise of the therapsids; amphibians increasing in size after making it onto the land; all animals increasing in size after the Cambrian Explosion 540 million years ago... and so on.
Greater size gives evolutionary advantages to both individuals and species for a number of reasons, the most superficial being that a larger beast is harder to kill and can kill or defend oneself more easily. In fact, larger size may be so evolutionarily beneficial that the only brakes on Cope's rule are that clades composed of larger individuals are more likely to go extinct in times of trouble (mostly because they have greater metabolic requirements than the small). However, there are other limiting factors -- for instance, biomechanics -- a bird that weighs too much cannot fly.
Cope's rule has been carried to the extreme in animals like whales, which evolved from land animals the size of wolves, and the extinct sauropods, dinosaurs that approached 200 ft (60 m) in length. There are many other examples, especially occurring in the wake of mass extinctions where all the larger animals are wiped out. Some scientists have proposed that Cope's rule could be an artifact caused by the fact that larger bones more readily fossilize, but the phenomenon seems sufficiently robust that this selection effect does not significantly diminish it.
Cope's rule unfolded most strongly in the Age of the Dinosaurs, when the average terrestrial vertebrate was much larger than those typical today. This may have something to do with climate, as the world was a much warmer, life-friendly place back then, while today it is in the midst of an extended Ice Age, with glacial maxima that cover much of the Northern Hemisphere and all of Antarctica in ice caps. This decreases the total amount of food available and makes what was once lush forests into chilly grasslands. In this environment, size can still be an advantage however, as it leads to warmth. This can be seen in the evolution of the woolly mammoth.