I was told that if you do the mathematical analysis and use a Forceault tester, a spherical mirror at f/8 will produce as sharp an image as a paraboloid.

Not necessarily at f/6 but at f/8 or longer. True?

At f/8, a spherical mirror will be "diffraction limited", with roughly 1/4 wave of undercorrection. But a spherical mirror doesn't stand a chance against good parabolic mirror at f/8.

Many good scopes have spherical mirrors and operate at f/8. Such as a few Tal scopes, and the Orion XT4.5. Also, in smaller scopes whether or not you have a parabolic mirror doesn't really matter. An 80mm f/6 with a spherical mirror is perfectly forgivable, and I've never heard of a 4.5'' f/8 with a parabolic mirror.

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You can make any mirror have 1/4 wave accuracy; this is independent of its shape. The problem with spherical mirrors, as I'm sure you know, is that they can't hold as sharp a focus as parabolic ones can due to spherical aberration. An f/8 parabolic mirror will hold better focus than an f/8 spherical; this is in fact true at any focal ratio, since it has to do with the geometry of the shapes. However, the eye also has limited resolving power, and can't tell the difference between a well-made spherical f/8 or higher and a parabolic f/8 or higher, even though the parabolic mirror still holds a better focus.

It depends on both f ratio and diameter. As mirror size goes up f ratio goes up as well. See this link for more detail.
Making your own telescope mirror

Rick

The eye has plenty of resolving power to resolve the diffraction pattern at high magnification. I have seen it myself with a 6-inch scope at 200x or so.

Skilled observers, using high magnification under perfect seeing, have reported visible differences between a perfect mirror and one with a 1/4 wave error. Sky and Telescope reported on just such a test with 6-inch f/8 mirrors in the March 1992 issue. The objects in question were low-contrast features on Jupiter, which are very unforgiving of even a slight reduction in contrast at or near the diffraction limit. The improvement with the perfect mirror was subtle, however, and average seeing masked it. For most observers under typical conditions, the 1/4 wave mirror was diffraction limited or nearly so. This is indeed the amount of spherical aberration in a 6-inch f/8 system.

Yes I should have said it would be less of a problem at f/6 and f/4 than at longer f ratios like f/10 or f/12.

When people say "Fast", does that refer to f/4.5 or f/12 ?

No, a spherical mirror is MORE of a problem at f/6 or f/4 than f/10 or f/12.

Fast refers to shorter focal ratios.

Usually scopes under f/6 are considered fast. F/6 to f/10 are considered medium focal ratios, and f/11 and beyond are slow. "Fast" or "Slow" are photographic terms, they refer to the shutter speeds needed to take a picture at a given focal length with lenses that are at different apertures. It is confusing to think of it with telescopes, so try not to. Fast focal ratios in telescopes only matter if you want to do astrophotograpy, or you are interested in getting the widest field possible in an eyepiece. But "slower" scopes are better planetary scopes unless you have a really well made "fast" scope, because optical aberrations are more evident in faster focal ratios. I myself am a "fast" scope guy, my main scopes are below f/5.

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First, although the terms 'fast' and 'slow' may have originally referred to photographic terms, if one thinks about the angle of the point of focus relative to the objective, 'fast' mirrors (or lenses) have sharper, steeper slopes. Slow ones do not and are very long.

I read in an article in Sky and Telescope a few years ago about an ATM guy who built a very long, folded spherical mirrored telescope. In the article, it stated that spherical mirrors could not perform up to parabolic standards till about F/17 or F/18, which is what his telescope was. And it was one big scope.

Given that MCT's are often F/13 and higher and must employ a thick, aspherical correcting lens, I would have to think that any reflector below F/15 would require a parabolic mirror to achieve diffraction limited performance, or else have to employ some type of corrector, if not an MCT, then something like the Vixen model.

Let's not get carried away comparing simple apples with compound oranges.

What was his aperture?
Is that Maksutov-Cassegrain? If so, it consists of a spherical primary of F/3 or so, a meniscus lens with spherical surfaces, and a convex secondary mirror in the form of an aluminized spot on the back surface of the lens. It does not follow from the need for the lens that such a lens is needed for a spherical 6-inch F/13 Newtonian. As has been pointed out already, an F/8 comes close to diffraction limited performance, and the error decreases rapidly with increasing focal length at the same aperture. Larger apertures are less forgiving.