The simple magnifying glass was in use in the thirteenth century but it was another three hundred years before it was used extensively and two or more lenses were used together to give a compound microscope.
A magnifying glass or microscope is an instrument used for looking at objects close to the eye and the image is usually formed at the least distance of distinct vision (D) from the eye.
This is also called the near point.
I' h' Figure 1
I

D
O' h
O
 u
You can see from Figure 1 that at this distance the image will subtend the largest possible angle at the eye and yet still remain clearly visible without strain.
At the near point the angular magnification of a simple microscope is defined as follows:
Angular magnification (M) =
 where

is the angle subtended at the eye by the image at the near point and

is the angle subtended at the unaided eye by the object at the near point.
.
A convex lens of focal length produces a magnified, erect and virtual image (
II
') of the object
(OO').
The lens is adjusted so that the image is at the near point and so the image distance is the least distance of distinct vision D, about 25 cm. From the diagram:
Angular magnification (M) =

/

= h'/h as long as the angles

and

are small.
However, the linear magnification of the lens is h'/h = v/u, and since 1/u + 1/v = 1/f we have that: v/u = v/f - 1 but also v = D
Therefore:
Angular magnification (M) = D/f - 1