MISCELLANEA. 
I. On some Dangers of Extrapolation. 
By EMILY PERRIN, University College, London. 
(1) It is well known that most excellent graduation results both in the case of physical and 
vital investigations may be obtained by fitting algebraic or trigonometrical curves to a limited 
range of observation. Such curves are in constant use for interjiolation, not only in physics and 
demography, but also in chemistry, in actuarial science, and in empirical investigations in all 
branches of technical research. So long as such curves or algebraic formulae are applied for the 
purposes of interpolation within the range of observation, their use is not only justifiable, but 
indeed indispensable. This statement of course supposes that a good formula has been selected 
and has been properly fitted to the observations. 
When we examine, however, the excellence of fit of even an arbitrarily chosen curve to a 
given range of data, when we see how it smooths the irregularities, and provides well within the 
limits of probable error the graduation that the chemist or engineer or actuary is eagerly seeking, 
we are tempted to continue our empirical representation of the data beyond the range of 
actual observation and predict the future from the past, or the condition of the unobserved from 
the observed. This extrapolation is one of the most fascinating fields of enquiry and yet one 
in which definite conclusions are most difticult to reach and pitfalls frequent and dangerous. 
How often is this extrapolation indulged in almost unconsciously by the scientific enquirer ! We 
know now that the proiiortionality of stress and strain is only true within narrow limits, yet the 
early investigators extrapolated from this linearity all across the mysteries of set, yield-point, and 
stricture, tip to rupture! Within quite recent j'ears we have, we fancy, seen distinguished 
physicists and chemists running along tangents as if every observation-curve beyond the observed 
range ended in an indefinitely extended point of inflexion. Yet their want of logic is only com- 
parable with that of the early physicists who extended Hooke's Law or Boyle's Law beyond the 
limits of obseiTation, or of the later physicists who apply the deductions from Hooke's Law to 
the stress and strain of the solid earth, or relations between physical attributes at known tem- 
peratures and pressures to what may be supposed to happen beyond the range at which it has 
hitherto been possible to experiment. 
The present enquiry is, of coui'se, of a much humbler nature, but it may serve possibly as 
a warning on the dangers of extrapolation. Even if it does not deal with such important pro- 
blems as the physics of the solid earth, it does deal with things which have been subjected to 
a like treatment. A parabola of a sufliciently high order — such a curve as has hitherto been 
often used by physicists and chemists — has been selected to describe the observed range of 
measurements, and then extrapolation has been attempted at a very small distance outside that 
range and the results compared with actual ol)servation. When the conclusion to be drawn is 
merely a note of wai'ning, may we not learn from the lesser as to the greater 1 It seems to us 
that such " extrapolation " at least is justifiable. 
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