616 VIII. ON THE STRUCTURE OF MATHEMATICS 







two points. i In referring our geometrical entities to co
ordinate axes, or frames of reference, as they are called, we are interested in the properties of our geometrical entities and not in the accidental characteristics of our frames of reference, or the accidental characteristics of the form of representation we are using. Mathematicians discovered long ago that the form of representation is not of indifference to the results they obtain. Speaking roughly they have discovered that in one form of representation, they obtained characteristics a, b, c, d,. . . m, n; in another form, characteristics a, b, c, d, .. . p, q; and in still another form, characteristics a, b, c, d, . . . s, t, . In cases where direct inspection was possible they find by checking up predicted characteristics, that such characteristics as a, b, c, d in our example actually belong to the subject of our analysis, whereas the characteristics m, n,... p, q,... s, t,.. . , do not belong to our subject at all, but vary from one form to another depending on the form of representation. Such facts make mathematicians distinguish between characteristics which are intrinsic, which actually belong to the subject independently of the form of representation; and those which are extrinsic, which do not belong to the subject, but are accidental and vary with the form of representation we happen to use.
If we mix intrinsic and extrinsic characteristics we have a structurally distorted knowledge of our subject. Obviously we are interested in methods by which these two types of characteristics can be separated and distinguished.
Such methods are found in what we call the transformation of coordinates, which means the passing from one form of representation to another, from one system of coordinates to another which corresponds to translation from one language to another. Obviously those characteristics which are intrinsic to our subject are and must be independent of the accidental selection of our form of

