Which way should I move my sight?

A fundamental assumption in moving the sight is that the next shot will go to the same place as the last if no adjustment is made.  If all arrows are the same and shot identically this is a fairly safe assumption.  A more likely situation is that several arrows need to be shot in a group before confidence can be assured that it is a sight setting problem that exists rather than just a defect in form causing the spread.

In this diagram the eye is seen to line up the sight ring with the centre of the target. The wavy line represents the (rather unlikely) flight path of the arrow from its initial position just below the eye to a position just above the aiming point. If every arrow follows this path we would simply move the arm down, leaving the sight where it is. Relative to the bow, this is the same as moving the sight up. This leads to the well known rule "move the sight to follow the arrows".

However it is possible to do better than that! It is clear from the diagram that the two triangles with the eye as apex are similar in a mathematical sense. Hence, if the error at the target is b and the target is at a range R then the distance the sight at a distance d from the eye has to be moved is b*d/R. For most practical purposes we may assume the ratio R/d = R so that the amount the sight has to be moved is roughly b/R. Knowing the distance between the rings of the target we can easily calculate the distance to be moved. For example, if the group is centred on the blue, the average error is five rings from the middle; on a 122cm face this is about 30cm. For a target at 60yd the sight has to be moved about 30cm/60 or 5mm. Of course the more precise formula will give a better answer but the distance to the eye is needed and a calculator required to do justice to the calculation. This refinement is hardly practical or justified until the group size is relatively small. At such a stage, the archer will have a better technique as well as a reliable set of sight marks!!!