How problems with telescopes lead to solutions
Chromatic aberration in refractors
It wasn't Galileo who invented the telescope - he just took credit for it. In fact, the refractors we sell in the shop aren't like Galileo's refractor at all - unless you are buying a pair of opera glasses (https://www.opticscentral.com.au/binoculars/opera-theatre-binoculars.html).
Galileo's refractor had a single convex lens out the front and a small concave lens for an eyepiece. It showed the image the right way up, but the field of vision was very small - it was like looking through a straw.
It was Kepler who replaced the concave lens at the back with a convex lens, lengthening the scope but giving a much bigger field of view. It didn't really matter if the image was upside-down because it was meant for astronomy and up and down don't mean much up there.
The problem with the Keplerian refractor was chromatic aberration. Blue light bends more than red light (with green light somewhere in between). This meant that the white stars looked like round rainbows - with blue haloes and maybe a green ring inside that and a red dot in the middle. Not all that flash.
What was needed was a genius. Yes, Isaac Newton, of course. He realised that glass lenses had "dispersion" and that a front-silvered mirror (his was made of polished metal) would avoid the problem. Newton polished his mirror into a spherical shape and built a reflecting telescope that showed vastly improved colours.
These days we can get around the problem in a different way. We use multiple objective lenses. Combining different shaped lenses with different refractive qualities we can design a lens group that refracts red, green and blue light all at the same spot.
Improvements to Newton's telescope
I've shown you diagrams of telescopes designed by Galileo (actually not - he pinched the design), Kepler and Newton, talked about the shortcomings of these, and how they overcame them over time.
Newton's reflecting telescope was designed to get around the problem of chromatic aberration. It did this by avoiding refraction entirely and replacing the glass lens with a front-silvered mirror.
The problem with this telescope was that the spherical mirror produced a different kind of aberration. Perhaps not surprisingly it is known as spherical aberration. Here, the light rays hitting the outer edge of the mirror get reflected to a point closer to the mirror than rays hitting the inner parts of the mirror.
Newton might have been pleased that there weren't colour fringes around bright stars, but wouldn't have appreciated not being able to get things into crisp focus.
In 1721, John Hadley gave Newton's reflector a parabolic mirror. Perhaps we should be calling our Newts "Hadlerian reflectors". Decent modern Newts now have parabolic mirrors.
Incidentally, even now we see cheap knock-off reflectors with spherical mirrors coming for collimation. There's not a lot we can do with these, apart from sympathise with Isaac Newton about how they're a pile of garbage.
However, even with Hadley's parabolic mirror, there's still something called off-axis aberration (typically called "coma"), where the stars look like little comets.
Today, we correct coma with a lens with the amazing name of a "coma corrector" (gasp!). They're meant for fast Newts (say, f/4 or faster).
Another modern fix for the aberrations of Newton's reflector (the one with the spherical mirror) is to introduce a Schmidt corrector plate. This is the nearly-flat glass element you see at the front of a Schmidt-Cassegrain. They're sold as a Schmidt-Newtonian telescope.
Getting more and more magnification
Last time we saw how a couple of different aberrations of Newton's reflecting telescope were ironed out. But they still didn't have the focal lengths that contemporary refractors did. Newton's original telescope only had a focal length of 165mm - the length of a modern guide scope.
Scientists needed more magnification, and a priest by the name of Laurent Cassegrain came up with a design where light first hit a heavily concave mirror and then a second convex mirror. The secondary nearly - but not quite - undid the work of the first, resulting in a very long effective focal length. You can see it in my scribbled ray diagram.
All the Cassegrain variants I've sketched here look similar - the parallel light rays get bent a lot, and then nearly - but not quite - straightened out again.
Cassegrain's scope had spherical mirrors - they're easy to grind like that. So, of course, it suffered from spherical aberration.
To counter this, Bernhard Schmidt developed a complicated corrector plate. This is a nearly-flat lens that is matched to a spherical primary mirror. In modern Schmidt-Cassegrains, the secondary mirror is mounted in the middle of the corrector plate. With all this, only a little coma remains. Also typically, matching corrector plates to primary mirrors is pretty expensive.
As a response, Dmitri Maksutov found a way of bending a corrector plate into a meniscus shape. This has a spherical front and a spherical back. The primary mirror is also spherical, and the secondary is silvered onto the back of the spherical corrector plate. This scope pretty much has it all - a long focal length, low cost and low aberrations. No wonder Maksutov was made a Hero of the Soviet Union.
One last scope design is the Ritchey-Chréchean, with hyperbolic primary and secondary. The only thing I have to say about this is that the difficulty in pronouncing the name is second only to collimating the damned thing.
Any questions or corrections, please put them in the comments below!