Compound Microscope: Magnifying Power

A compound microscope uses two convex lenses in place of the simple microscope's single lens, to reach far higher magnifying powers than a single short-focal-length lens safely can. The lens nearer the (very small) object, called the objective, has a very short focal length $f_o$ and is positioned just beyond its own focus from the object, so that it forms a real, inverted, and already substantially magnified image of the object a little way inside the instrument's tube. This real intermediate image then acts as the object for the second lens, the eyepiece, which has a somewhat longer focal length $f_e$ and is positioned so that this intermediate image falls just inside the eyepiece's own focus, so that the eyepiece works exactly like a simple microscope, taking the already-magnified intermediate image and magnifying it a second time into a large final virtual image viewed by the eye. The overall magnifying power of the compound microscope is the product of the two individual magnifications -- the objective's own linear (lateral) magnification $m_o$ of the intermediate image, multiplied by the eyepiece's own angular magnifying power $M_e$ acting on that intermediate image as its object: $M=m_o\times M_e.$ When the final image is adjusted to form at the eye's near point $D$ (the normal way a compound microscope is used), and writing $L$ for the distance between the objective's own focus and the eyepiece's own focus (very nearly equal, for a well-designed instrument, to the tube length -- the distance between the two lenses), this combination works out to the commonly-quoted approximate formula $M\approx\frac{L}{f_o}\left(1+\frac{D}{f_e}\right).$ Because $L$ is typically many times larger than either $f_o$ or $f_ …