[StephenBatey]

I won't frighten by using an equation, but there is a link fixed by the laws of optics that relates the focal length of a lens, the distance of the subject and the distance behind the lens that a sharp image of said subject will be focused. As the subject gets closer, the sharp image gets further away, assuming that the focal length doesn't change..

Before autofocus, the best lenses focused by moving them away from the camera body in a focusing mount or by using bellows. As the subject got closer, you needed more distance, and eventually the focusing mount built in to the lens was insufficient. You then used tubes or bellows to increase the separation. Hence what they do is let you focus closer.

After autofocus, lens designers began to cheat a little to make the autofocus faster and less energy expensive by focusing by altering the focal length of the lens so that the lens/image distance didn't have to change much and therefore the lens didn't have to be physically racked out.. But there is still a limit as to how much you can change this.

Close up lenses work differently. When the subject is one focal length in front of a lens, the in focus image is located at infinity. Hence, put on a close up lens, and the camera lens can be set at infinity and a subject one focal length (of the close up lens) will be in focus.

Close up lenses are usually marked with their focal length expressed in diopters, 1 diopter being the reciprocal of the focal length in meters. The reason for this odd convention is that when you stack them, the diopters are additive - a 1 diopter lens lets you focus on a subject 1 meter away; add another and as 1 + 1 = 2, the subject can now be 1/2 meter away and be sharp.