An n-dimensional Weber Law and the Corresponding Fechner Law.

Weber's law of 1834, DeltaS/S=c for the just noticeable difference (jnd), can be written as S+DeltaS=kS, k=1+c. It follows that the stimulus decrement required to elicit one jnd of sensation is S-DeltaS*=k(-1)S. If generalized for two stimulus dimensions and two corresponding response dimensions, Weber's law would have to state such equations for all directions of change in the plane. A two-dimensional Weber law with exactly these properties is realized by [S(x)+DeltaS(x)(straight theta), S(y)+DeltaS(y)(straight theta)]=[k(sin(straight theta))S(x), k(cos(straight theta))S(y)] which determines the stimulus coordinates for all stimuli just noticeably different from the stimulus (S(x), S(y)) in all directions 0</=straight theta</=2pi. Fechner's problem now is understood as finding a transformation of the plane which maps the set of stimuli one jnd apart from the standard stimulus onto a unit circle around the standard stimulus' image. This transformation (R(2)(+)-->R(2)) is [x, y]mapsto[log(k)(x), log(k)(y)]. The solution is generalized to arbitrarily many dimensions by substituting the sin and cos in the generalized Weber law by the standard coordinates of a unit vector. Copyright 2000 Academic Press.