# Topic 41. The Galilean telescope

The Galilean telescope was first invented by Galileo in about 1609. Unlike the astronomical telescope, which is made using two positive lenses separated by the sum of their two focal distances, the Galilean telescope has a positive objective and a negative eye lens, which are separated by the difference between their two focal distances.  As in the previous discussion, we consider an object at a very great distance, D, from the eye. The magnification of the eye itself is d/D, where d is the distance between the eye-lens and the retina.

The objective lens, whose focal length is f, performs the same function as in an astronomical telescope: acting alone, it would form a real, inverted image of the distant object, as shown in the following diagram:

The second negative lens is positioned so that its second focal point is exactly coincident with the second focal point of the objective lens. Thus the rays that are aimed at producing the real, inverted image shown above, actually hit the second lens before they get to the image.

We can use the standard ray tracing rules to construct the image produced by the second lens, keeping in mind that the top ray in the figure above, which was headed for the second focal point of the first lens, is also headed for the same focal point of the second lens, since the two are coincident by construction. Using the ray-tracing rules for a negative lens, this ray is refracted parallel to the axis, and this is shown in red.

The dark green ray is another ray that forms the image produced by the first lens. Its direction is known from the figure above, since it originates at the object, is refracted by the first lens and ends at the image of the first lens. Note that it is also drawn to pass through the center of the second negative lens. The second lens has no effect on this ray, since it passes through its center.

The dark green ray and the red ray appear to come from the purple image as shown. Therefore, the effect of the telescope is to produce a virtual, erect image of the distant object. Note that the image is erect rather than inverted. This virtual image is then viewed by the lens in the eye to form a real image on the retina.

Using the algebraic form of the ray-tracing rules, we can show that the magnification of this arrangement (relative to the unaided eye) is the same as for an astronomical telescope whose lenses have the same numerical focal lengths.

This arrangement is smaller than an astronomical telescope of the same magnification, since the lenses are closer together. In addition, the image is erect and so does not need a separate inversion as with the astronomical design. This configuration is often used in “opera glasses” and other simple binoculars where small size is an important requirement.