We're used to science being a discipline of professionals using expensive equipment and working for large universities, organizations, or even corporations. It wasn't always like that of course. Science used to be the domain of the talented, and more than occasionally self funded, amateurs.
It turns out there is still a place for amateurs in science. Especially in astronomy. Average people are able to look into the night sky and observe much of the wonders around us. Maybe not with the detail and precision of a Hubble space telescope or a large earthbound observatory. But there's more than enough up there to look at.
In fact the amateurs have been more successful than the 'professionals' in finding new comets in our solar system. Martin McKenna wrote an Visual Comet Hunting - A Deeper Look. In it he writes about the various 'professional' systems that may end up being better than amateurs and pretty much concluded that the amateurs have nothing to worry about for the time being. They're still likely to find new objects before the new systems.
Even looking at known objects can be incredibly rewarding. Sites like SkyChasers.net and club sites like The Hamilton Amateur Astronomers are filled with galleries of pictures taken by non-professionals.
You can go and buy a good set of binoculars or a telescope to view the heavens but for an added challenge you can also make your own. Amateur telescope making has a long history and there are a number of resources available. The ATM Site (Amateur Telescope Makers) has lots of articles on many aspects of making your own telescopes. From notes on silvering and coating your mirrors to some thoughts about spiders in Newtonian telescopes. And no... that second one isn't about getting rid of a bunch of arachnids... it's about the thin supports, or spider, that hold the little mirror at one end of a Newtonian telescope.
Of course if you are going to design and build your own telescope it may help to know the physics and the math involved. Getting the best image, or knowing how good an image you can get, is complicated business. Vladimir Sacek's Notes on Amateur Telescope Optics site is as close to a complete textbook on the subject as you can find. Heavy on the math, detail, and diagrams it is dense but if you're trying to understand and work out telescope design there is no better place to start.
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