102 
RADIATIVE EQUILIBRIUM 
is to obtain “insight”—to see which of the numerous factors are par 
ticularly concerned in any effect and how they work together to give it. 
For this purpose a legitimate approximation is not just an unavoidable 
evil; it is a discernment that certain factors—certain complications of 
the problem—do not contribute appreciably to the result. We satisfy 
ourselves that they may be left aside; and the mechanism stands out 
more clearly, freed from these irrelevancies. This discernment is only a 
continuation of a task begun by the physicist before the mathematical 
premises of the problem could be stated; for in any natural problem the 
actual conditions are of extreme complexity and the first step is to select 
those which have an essential influence on the result—in short, to get 
hold of the right end of the stick. The correct use of this insight, whether 
before or after the mathematical problem has been formulated, is a faculty 
to be cultivated, not a vicious propensity to be hidden from the public 
eye. Needless to say the physicist must if challenged be prepared to 
defend the use of his discernment; but unless the defence involves some 
subtle point of difficulty it may well be left until the challenge is made. 
I suppose that the same kind of insight is useful to the mathematician 
as a tool; but he is careful to efface the tool marks from his finished 
products—his proofs. He is content with a rigorous but unilluminating 
demonstration that certain results follow from his premises, and he does 
not generally realise that the physicist demands something more than this. 
For the physicist has always to bear in mind a thousand and one other 
factors in the natural problem not formulated in the mathematical problem, 
and it is only by a demonstration which keeps in view the relative import 
ance of the contributing causes that he can see whether he has been justified 
in neglecting these. As regards rigour, the physicist may well take risks 
in a mathematical deduction if these are no greater than the risks incurred 
in the mathematical formulation. As regards accuracy, the retention 
of absurdly minute terms in a physical equation is as clumsy in his eyes 
as the use of an extravagant number of decimal places in arithmetical 
computation. 
Having said this much on the one side we may turn to appreciate the 
luxury of a rigorous mathematical proof. If the results obtained do not 
agree with observation the fault must assuredly lie with the premises 
assumed. The mathematician’s power of narrowing down the possibilities 
supplements the physicist’s power of picking out the probabilities. If 
space were unlimited we might try to duplicate investigations where 
necessary so as to satisfy both parties. But if one investigation must 
suffice I do not think we should usually give way to the mathematician. 
Cases could be cited where physicists have been led astray through in 
attention to mathematical rigour; but these are rare compared with the 
mathematicians’ misadventures through lack of physical insight.