Step 1: Set-up of the instrument.
Two converging lenses share a common axis inside a tube. The front lens (objective, focal length \(f_o\)) is small and close to the tiny object; the back lens (eyepiece, focal length \(f_e\)) is next to the eye. The object \(AB\) sits just outside the objective's focus.
Step 2: What each lens does (ray picture).
Rays from \(AB\) refract through the objective and actually cross to form a real, inverted, enlarged image \(A'B'\) within the tube, positioned just inside the focus of the eyepiece. The eyepiece then treats \(A'B'\) as its object and, like a magnifying glass, sends out diverging rays whose backward projections form a much larger virtual image \(A''B''\) at the near point \(D\).
Step 3: Split the magnification into two factors.
Total angular magnification \(M = m_o\, m_e\). The objective gives linear magnification \(m_o = v_o/u_o \approx L/f_o\), where \(L\) is the tube length. The eyepiece, used as a simple microscope with image at \(D\), gives \(m_e = 1 + D/f_e\).
Step 4: Combine.
Multiplying the two factors,
\(M = \dfrac{L}{f_o}\left(1 + \dfrac{D}{f_e}\right)\).
For a relaxed eye (final image at infinity) the eyepiece factor becomes \(D/f_e\), giving \(M = \dfrac{L\,D}{f_o f_e}\). Because both \(f_o\) and \(f_e\) are kept small, \(M\) is large.
\[\boxed{\ M = \dfrac{L}{f_o}\left(1+\dfrac{D}{f_e}\right);\quad M=\dfrac{LD}{f_o f_e}\ \text{at infinity}\ }\]