Summary:  Common optical systems

A simple model of the human eye is an optical system with a fixed image distance x_i and a single lens with an adjustable focal length f.

For the normal eye the equation 1/x_i + 1/x_o= 1/f can be satisfied for object distances x_o from infinity to 25 cm by adjusting f.

You get the highest angular magnification (the biggest image) if you put your normal eye 25 cm from the object.

A magnifying glass can improve this angular magnification by a factor
MP = (25 cm)/f_mag, if you put the magnifying glass a distance f_mag from the object.

A telescope is not an image forming system by itself, but it provides light gathering ability and angular magnification for another optical system, such as the eye or a camera.  Make sure you understand the design differences between a Keplerian, a Gallilean, a terrestrial, and a reflector telescopes.  The matrix element M12 of all telescopic systems is zero, the telescope does not have a finite focal length.

A microscope is an image forming system with an objective lens of focal length f_o and a fixed image distance f_o + g, followed by a magnifying glass (the eyepiece).
Here g is called the tube length of the microscope, and –g/f_o is the linear magnification of the objective.  The eyepiece provides an additional angular magnification M_eyepiece, so that that the magnifying power of the microscope is given by MP = -(g/f_o)MP_eyepiece.