Two Simple Telescopes and Microscopes
The two simplest refractors use two lenses each. An inverted image is fine for looking at stars, but if you want an upright image, use the Galilean telescope.
Building an Astronomical Telescope
Use two convex lenses with focal lengths fo and fe, objective and eyepiece. Ideally the eyepiece should have a small focal length and objective large, because it can be shown that the angle magnification is
M = -fo/fe
(Look at the bottom of some object, then the top. Your eyes rotated upward an angle q . Then repeat, using an inverting telescope and looking at the image of the object. Your eyes rotated downward (so consider it negative) angle q '. M is q '/q .)
Make the distance between the lenses fo + fe, but allow the distance to be adjustable to allow for individual differences in eyesight.
Building a Galilean Telescope
Use a concave lens for the eyepiece (negative focal length) and make sure that fo is greater than the absolute value of fe. It turns out that the above relationships are valid for this telescope. M=-fo/fe and L=fo+fe and be sure to use the negative, so L is less than fo.
(Incidentally, the one-way viewers in some hotel doors are convex-concave combinations. The convex lens is on the inside, and it magnifies the virtual image of the concave lens a little, but its main purpose is to prevent the person in the hall from seeing a clear image of things in the room.)
Turning them into microscopes:
Just lengthen them significantly. Then keep the distance between lenses fixed and vary the distance from the object to the objective lens to get the best view of the object. It turns out that the magnification is approximately
M = NL/(fofe), where N is the "near point" distance, the distance from your eye to a tiny object when you are trying to see best it without magnification.
Note that since fo is in the numerator of Mtelescope and in the denominator of Mmicroscope, a high magnification telescope turns into a low magnification microscope and vice-versa, other things being equal. Of course sometimes low magnification is desirable. You may not be able to see the whole thing at once with too much magnification.
Comments, questions: email fredrick.gram @tri-c.edu (remove space before @)