The Apparent Size of an Object 
When an object is observed, the light entering the eye forms a
real (and therefore inverted) image on the retina, the "screen" at
the back of the eye. 
Without the need for a detailed ray diagram, we can predict the
size of the image on the retina as shown below, simply
remembering that a ray through the optical center of a lens
(or system of lenses) passes through undeviated. 



The size of the image on the retina of the observer depends on
two rather obvious factors: 
1. the real size of the object, h 
2. the distance of the object from the observer, u. 

However, as also shown in the diagrams above, these two factors
together determine the size of the angle subtended by the object
at the eye, the angle a in the
diagrams. 
In other words, we can say that the perceived size of
the object will depend on the angle
α. 

The apparent size of an object is directly proportional to
this angle. 

Optical instruments like microscopes and telescopes provide
magnification by increasing the size of the angle subtended at the
eye. 

Angular Magnification by a Simple Microscope 
The angular magnification (sometimes called magnifying power) is
defined as follows 



In order to see an object in as much detail as possible the
person would be expected to hold it as close to the eye as
possible. 
The closest that a person can hold an
object and still see it clearly is called the least distance of
distinct vision, D. 
An object placed at this distance from the eye is said to be at
the near point of the eye. 
This distance varies from person to person but is taken to be 25cm
on average. 

Consider first an observer looking at a small object, placed at
the near point. 

Assuming α is a small angle
then 

Now consider the person to be using a simple microscope, with
the object distance, u, chosen to produce an image at distance D
from the lens. 
If the eye is placed close to the lens, the angle subtended at
the eye by the image is approximately given by 


So the angular magnification, in this situation, is given by 

Therefore, when the image distance is D, the angular
magnification is (not surprisingly) equal to the linear
magnification. 

Rearranging the lens equation gives 

In this case, M = v/u and v = D, therefore 


Equation 1 

for a simple microscope with the image at the near
point of the observer. 

We will now consider the situation in which
the object is moved so that the image
is at infinity. 
Obviously, as the object distance has
increased, β is now a smaller angle
than in the previous case. 

Again, remember that β is a
small angle, so we can write 

so the angular magnification is now given by 

which means that 


Equation 2 
If we compare equations 1 and 2 we might, at first sight, be tempted
to think there's a mistake somewhere... 
The magnification given by equation 2 must be less than that given by
equation 1. 
If you can't see why this is the case... see here. 