The single-lens telescope

In my first post I mentioned that Roger Bacon has written, in his Opus Majus, of  the use of lenses to magnify distant objects.  He does not go into any detail but it is indeed possible that he did observe that  a single convex lens of sufficiently long focal length before his eye would make remote objects appear nearer.  In the figure below we represent  the observation of the image of a distant object formed by a convex lens at its focus.

Fig.1 

Write $\alpha$ for the angle subtended by the image at the eye, and $\beta$ for the angle subtended by the object at the lens.  Since rays through the optical centre of a thin lens are undeviated, $\beta$ is also the angle subtended by the image at the lens. Furthermore, if the object is taken to be effectively at infinity $\beta$ is the angle subtended by the object at the eye.  We now readily see that the angular magnification is
\[ m=\frac{\alpha}{\beta}= \frac{f}{d} \]
where f is the focal length  IL  of the lens, and EI to the the least distance of distinct vision d . 
Thus taking d as the conventional value 10″, a focal length of 20″ would give us a magnification of two.

If it had been recognised that the above shows the true state of affairs, that the lens produces a real image, hanging in space, as it were, then it would have been a small step to take a second lens of small focal length to examine and magnify that image and the telescope might have been born much earlier than it was.  That state of affairs was unrecognised until after Kepler, but that is another story.


In 1585, in a memoir on optics written at the request of Lord Burghley, William Bourne described glasses for improving both near and distant vision,   Henry C King, in his  The History of the Telescope (1955)  comments that Bourne's eyes were undoubtedly hypermetropic (for which, see the previous two posts) since as he moved his eye further and further back from the focus of the lens, distant objects became "of marvellous bignesse".  With his eye at the focus nothing could be seen, while nearer the lens he could see the distant object 'reversed and turned the other way'.  

'Reversed and turned the other way' is what Bourne saw in the situation as shown in figure 1 above, with his eye accommodating to the image - how well, is not mentioned.  Why he was  also able to see images "of marvellous bignesse", by implication the right way round and the right way up, is depicted in figure 2.  

Fig.2

In the absence of the eye an image is formed by the lens at its focus, its focal length being IL=$f$. When the eye is interposed the image becomes a virtual object at a distance IE=$d$ behind the eye.  As we have seen in previous posts, since rays through the optical centre of a lens are undeviated, such a ray through any point on the object passes through the corresponding point (the  conjugate point) on the image.  Hence object and image subtend the same angle at the centre of the lens. It will be seen that if $\alpha$ is the angle subtended by the image at the eye, and $\beta$ the angle subtended by the object at the lens, then the angular magnification is 
\[ m=\frac{\alpha}{\beta}= \frac{f}{d} \]

the same expression as before.  We do not know what the values of $f$ and $d$ were in Bourne's case.