The Focal Length
Making a 4.25 Inch Dobsonian Reflector Telescope
Some Background, the F Value
The focal length of a mirror is the distance from the mirror to where an object at infinity will focus. This is usually measured in inches or centimeters. You will also run into the F value a lot. Typically F values are F/6, F/8 or F/10. This value has the same meaning as the F stop values on your expensive camera. The F value is measured by dividing the focal length by the diameter of the lens. So a 10 inch F/8 mirror has a focal length of 10 x 8 = 80 inches. This is almost 7 feet. While a Newtonian reflector will bend this path sideways, you can still expect the telescope tube to be this long.
Spherical and Paraboloid Surfaces
You may recall that when grinding an polishing a mirror, we are trying to mate two surfaces. This creates a spherical surface. But the type of surface we need for a telescope is a paraboloid. This is just the three dimensional surface you get by spinning a parabola around its axis. It's only when you get to the final stages of figuring the mirror that you change the surface to a paraboloid. This works because the difference in the two curves on the surface of the mirror is very small, just thousandths of an inch. There is no such thing as a mathematically perfect mirror, but a mirror that delivers light to where its going to an accuracy less than the wavelength of light is effectively perfect.
And that's where the F value comes in. For a 4.25 F/10 inch mirror the difference between a sphere and a paraboloid is not enough to keep the mirror from being optically perfect. You don't have to figure the mirror at all, just polish it into a sphere. For larger mirrors and smaller F values, the difference increases, and the figuring step becomes crucial.
If someone every tries to sell you a telescope based on its magnification, walk away. Any telescope can have just about any magnification you want. That doesn't mean you will want to look through it. The magnification you get from a telescope is approximately the ratio of the main objective's focal length, that's the mirror, and the focal length of the eyepiece. Larger focal length of the eyepiece, lower magnification. Smaller focal length of the eyepiece, higher magnification. Unfortunately there are many other things to consider. To begin with, higher magnification is not always a good thing. With high magnification it becomes increasingly hard to find a stellar object. Also, objects will move quickly across the field of view, forcing you to re-adjust the position frequently. That is unless you have a clock drive tracking the object. In addition, increasing the magnification on a small fuzzy object will generally only allow you to see a very large and extremely fuzzy object. So the bigger your mirror, the longer the focal length will be, and therefore the higher the magnification you will get. But if you want lower magnification you will need an increasingly larger focal length eyepiece. And large focal length eyepieces are more generally more expensive. The lens itself will have a larger diameter. This is a good thing, as it makes it easier to look into.
The Goal Here
The point of all this is how do you choosing the focal length of your mirror. It is the second most important choice you will have to make after deciding how big a mirror to use. In fact both choices should be made together. There are two issues to consider. First, what do you want to see with your scope and second, how much agony are you willing to deal with when trying to figure the curve. I'll suggest a few options.
Maybe you just want to look at the moon, the prominent planets; Jupiter, Saturn, Mars and Venus, as well as a few prominent and easy to find objects, eg. the great cluster in Hercules. In that case a 4 or 6 inch telescope will serve you well. An F/8 value is considered a good compromise. A larger F value will give you greater magnification, which if you read the section above, is not what you really want. A smaller F value and you will spend many many hours trying to get your mirror curve to be a paraboloid.
The Planet Watcher
The nice thing about planets is that they are bright. The compensation is that they are small. On a blurry night, you will see blurry planets, no matter how nice your telescope optics are. While you are not all that concerned about light gathering power, this is a catch with using a small telescope. What you want to see is details on the surface of the planet. That would be the great red spot and the bands around Jupiter. Saturn also has bands, as well as separately rings. You might also want to see canals on Mars, who knows. So to get both the resolving power and the magnification, you might want a medium size mirror with a somewhat longer focal length, say an F/10 8" mirror. That catch here is that this scope will have a fairly long tube. You will probably end up needed a step stool and maybe even a ladder. The good news is that figuring the mirror for this hard to transport scope will be straight forward.
Deep Sky Objects
To see many deep sky objects you want three things, light, light and light. To see nebulae, star clusters and galaxies, you want as much light as you can get. Ok, so like me you think, a 12.5 inch mirror would be nice. But to move around the F/10 version of this, you need a pickup truck. On top of that, having a long focal length increases the magnification and thereby narrowing the field of view. And for many deep sky objects you want the opposite, as wide a field as you can get. So you have to think about a fairly small F number, maybe F/6 or F/5 even. Any smaller and you will have difficult with the field flatness. That is, when you focus on the center of the eyepiece, the outer edges will be out of focus. The catch here is that figuring an F/5 12.5 inch scope could take more time than grinding and polishing it. It just has to be done carefully