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Firstly, types of telescopes are classified by the type of electromagnetic (EM) radiation they are used to observe. The most types of telescopes are optical telescopes, which observe the visible portion of the EM spectrum, followed by radio telescopes, infrared telescopes, gamma-ray telescopes, and x-ray telescopes. Only visible light and radio waves are easily observable from the Earth's surface — most other types of electromagnetic radiation are blocked by the Earth's atmosphere. This radiation must thus be largely observed using space-based types of telescopes, like the recently launched Spitzer Space Telescope, which is exclusively devoted to infrared light.
Although the basic technology necessary to build a telescope, polished lenses, have been widespread since the 11th century, it wasn't until 1608 that several Dutch lensmakers, most famously Hans Lippershey, made the first functional telescopes. These "Dutch perspective glasses" only magnified objects by a factor of three. Galileo Galilei overheard the basic concept while traveling in Venice in May 1609, and independently recreated the telescope, eventually coming up with a version that could magnify objects by a factor of 32. These early telescopes were refracting in nature, meaning they exclusively used lenses (a convex and a concave) to magnify the image and increase its apparent brightness.
Refracting telescopes were the first major types of telescope, remaining popular for hundreds of years. But there was a limit to how powerful they could be. These telescopes required larger and larger lenses to get more powerful, but making a lens of high quality becomes exponentially more difficult with size, due to tiny imperfections in the glass and "lens sagging" caused by the force of gravity acting on the lens.
In 1688, Isaac Newton created the first practical reflecting telescope, intended to solve the problem of chromatic aberration universal among the refractive telescopes of the time. The reflecting telescope introduced another specimen to the types of telescopes available for early scientists. Reflecting telescopes can be made much more powerful than refractive types of telescopes, as it is easier to make large, perfect mirrors than lenses. Some modern telescopes, called catadioptric telescopes, use a combination of mirrors and lenses to resolve objects, but reflecting telescopes are still the most common.