 As Stevens notes, scale types are individuated by the families of transformations they can undergo without loss of empirical information. Empirical relations represented on ratio scales, for example, are invariant under multiplication by a positive number, e.g., multiplication by 2.54 converts from inches to centimeters. Linear interval scales allow both multiplication by a positive number and a constant shift, e.g., the conversion from Celsius to Fahrenheit in accordance with the formula °C × 9/5 + 32 = °F. Ordinal scales admit of any transformation function as long as it is monotonic and increasing, and nominal scales admit of any one-to-one substitution. Absolute scales admit of no transformation other than identity. Stevens’ classification of scales was later generalized by Louis Narens (1981, 1985: Ch. 2) and Luce et al. (1990: Ch. 20) in terms of the homogeneity and uniqueness of the relevant transformation groups.