4 I.e. IXI, X2, x.1 are to be replaced by IXI *, x?, Xa *} when' XI * = allXI + al~X2 + a13Xa + bl, X2 * = a~lxl + a22X2 + auXa + b2, 'l:a* = aa.XI + aa2X2 + aa3Xa + ba, the a,j, b, being constants, and the matrix (a,j) what is known as orthogonal. 6 I.e. there is nothin~ in geometry to fix either origin or thc frame of reference when positions are concerned; and nothing to fix the frame of reference when vectors are concerned. 6 I.e. the b, = 0 in footnote 4 above. Sometimes a wider concept of matrices is those with det&rminants F O. We need not discuss these matters here. permissible,-all 'But no quantitatively reproducible method of thermometry.

· notion of utility when this is based on the conventional idea of prefer In these cases one may be tempted to take the view that the quantity ice. question is not numerical at all, considering how arbitrary the description numbers is. It seems to be preferable, however, to refrain from such alitative statements and to state instead objectively up to what system transformations the numerical description is determined. The case en the system consists of all monotone transformations is, of course, a her extreme one; various graduations at the other end of the scale are inhomogeneous or homo transformation systems mentioned above: eous orthogonal linear transformations in space, linear transformations one numerical variable, multiplication of that variable by a constant. fine, the case even occurs where the numerical description is absolutely orous, i.e. where no transformations at all need be tolerated. 2 3.4.6. Given a physical quantity,

e. with the stage of velopment of the subject. Thus temperature was originally a number ly up to any monotone transformation. a With the development of ermometry-particularly ansformations were restricted to the linear ones, Le. only the absolute ro and the absolute unit were missing. Subsequent developments of ermodynamics even fixed the absolute zero so that the transformation stem in thermodynamics consists only of the multiplication by constants. xamples could be multiplied but there seems to be no need to go into this bject further.

One may datum in this domain is the ke the attitude that In this case utilities are la.tion "greater," Le. the concept of preference. umerical up to a monotone transformation. This is, indeed, the generally ccepted standpoint in economic literature, best expressed in the technique f indifference curves.