226 ANNUAL EEPORT SMITHSONIAN INSTITUTION, 1930 



measure the temperature of the tape and the force used in holding 

 the tape taut, and then by means of the coefficient of expansion and 

 the coefficient of elasticity, apply corrections to the observed length. 

 It is hard to see what methods primitive man could have used in 

 applying such corrections to his distances measured by pacing. 



Why do we now apply such corrections? Merely because it has 

 been found by experiment that the result that we get by applying 

 such corrections is a quantity which proves to be more useful in 

 describing natural phenomena than the results we get without these 

 corrections. We must not think that we do it in order to obtain the 

 " true " or " absolute " length. 



To-day we have many other methods of measuring length than by 

 use of measuring rods or steel tape. For example, we use optical 

 instruments and measure distances by triangulation, we measure 

 heights in the atmosphere by means of a barometer, we measure the 

 distances of spiral nebulae by measuring the brightness of the 

 Cepheid variables observed in them by our most powerful telescopes, 

 we measure the lengths of molecules by finding the area of a water 

 surface over which a given amount of oil will spread, we calculate 

 the diameters of molecules by measurements of the viscosity of gases 

 by means of the kinetic theory, or we use X-ray diffraction patterns 

 or, finally, we calculate the diameter of an electron from its mass 

 and charge by means of the electromagnetic theory assuming that 

 all the energy of an electron lies in the electric field outside of its 

 surface. 



Now each of these measurements of length involves an entirely dif- 

 ferent set of operations and, therefore, fundamentally, according to 

 Bridgman, we should regard them as different concepts; logically, 

 in fact, they should all have different names. It has, however, been 

 found as a matter of experiment that two or more of these methods 

 when applied to the measurement of the same distance give results 

 which agree more or less with one another. This, then, is our justi- 

 fication for calling all these concepts by the same name, length. 



We may, if we wish, extrapolate and predict that by applying 

 suitable corrections to each of these methods of measuring lengths 

 we may be able to get better and better agreement between them. 

 Such methods of extrapolation may be useful and stimulating but 

 we must always expect that sooner or later we will be unable to 

 obtain agreement between these methods with more than a limited 

 degree of accuracy. This may not be due merely to experimental 

 difficulties but may often result from unavoidable fuzziness in the 

 concept itself. Such concepts as the diameter of a complicated mole- 

 cule, or the mean free path of a molecule in a gas are inherently fuzzy 

 conceptions and can mean not much more than when we speak of 



