Neutral data fitting by lines and planes

Given data on two or three variables we are interested in fitting a line or plane for the purposes of modelling the relationship between these variables. Most of the literature on this subject approaches this problem by selecting one of these variables and treating it differently from the others in the fitting procedure. This is acceptable if the purpose is to make predictions of that variable, but if we are seeking an underlying scientific law or a law-like relationship then it would seem more reasonable to treat all variables in the same way, unless there are particular reasons for not doing so. Thus in this paper we consider procedures for fitting lines or planes which treat all the variables equally.