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The basis of modern biological study is evolution. Like it or not, modern theory holds true that species change over time and the theory of natural selection currently is the best explanation we have for this phenomenon.
How do we study evolution?
We study evolution in many different ways, but it almost always boils down to cladistics, which is the way of determining the relationships between species in a classic family tree style. Ask any biologist where most of the data for the more extended family trees comes from, and they will most likely answer, 'the fossil record'.
Fossils tell us what the world was like millions, even billions, of years ago. They tell what flora and fauna lived where, if an area was a desert, plain, forest, lakebed, beach, river basin, delta, ocean floor, and so on. Even assumptions on the climate itself can be made.
But what is a fossil?
A fossil is any trace left over from ancient life that has, over the millennia, either been turned to stone or preserved in a similar way. These are not limited to teeth and bone, but also can be in the form of footprints, coprolites (fossilised dung), shells, leaves, impressions, and even petrified wood. Generally, all these types can be organised into two categories, cast and mould.
A cast fossil is a fossil that retains the shape of the original object. The dinosaur bones you see at a museum are examples of cast fossils. A mould fossil is like an impression. The original object is no longer there, but a mould of its shape is. Fossilised footprints and many plants are preserved as mould fossils.
How does a fossil form?
Basically, the specimen has to turn to stone. In the case of mould fossils, this means the impression has to remain intact long enough for the sediment to be compressed and heated into stone. Cast fossils including, but not limited to, teeth, bones, and wood fossilise when minerals permeate through the pores in the specimen and solidify. Amazingly enough, this means that portions of the tissue itself can be preserved locked in the rock.
Are fossils valuable?
Well, they are all valuable to palaeontologists (scientists who study fossils). If you are talking monetary value, it depends on the fossil. They range from the worthless to the priceless. Some fossils are of such importance, and so rare, that no price tag could ever be placed on them. The famous Berlin Specimen of Archaeopteryx is a prime example.
Ok, so some fossils are priceless, but why are some worthless?
In an economical sense, a fossil is worthless if there is a glut of them. The supply exceeds the demand. Many of the sea-shell fossils found in the Great Plains region of the United States are worthless. No one really wants them, and they can literally be picked off the ground.
What should I do if I find a fossil?
Depends on the fossil, and where it was found. Generally, you should ask the owner of the land if it's OK to hunt for fossils on their land. This is mainly common courtesy, but it can keep you out of trouble. It is actually illegal to remove fossils from some government-owned lands. If it is something small, like a shell, consider it a souvenir. If it is something somewhat larger, like a mastodon (a large elephant-like animal) or a hadrosaur (a large two-footed dinosaur), let a local museum or university know. If they are uninterested, you now have a big souvenir. Whatever you do, though, do not steal fossils, especially ones that scientists might find beneficial.