102 
Miscellanea 
But now let us look at the problem of extrapolation. Can we prophesy what happened in 
1887, the year before our range of observation starts, or in the years 1901 and 1902, the data of 
which have been received since our curves were determined? Table III. gives the actually 
observed and the predicted results, (i) on the basis of a straight line extrapolation, (ii) on the 
basis of a high parabolic extrapolation. 
TABLE III. 
Extrapolation Galmlations. 
Year 
Population ii 
lOOO's 
Death-Eatfe per 1000 
Still-born Eate 
per 1000 
1 Parabola 
Line 
Actual 
Parabola 
Line 
Actual 
Parabola 
Line 
Actual 
A 
A 
A 
A 
A 
A 
1887 
1901 
1903 
251 
! 935 
1162 
-.-187 
+ 87 
+ 284 
430 
848 
878 
-•08 
•00 
+ •08 
438 
848 
870 
28^38 
19^15 
20^39 
+ ^79 
+ ^55 
+ 4^19 
29^84 
15^24 
14^20 
+ 2^25 
-336 
-2-00 
27-59 
18^60 
16-20 
2-20 
1-73 
1-79 
+ -36 
+ 09 
+ 11 
£•66 
143 
134 
+ •82 
-•21 
-•34 
r84 
r64 
1-68 
Mean A 
Extrapolation 
186 
_ 
•05 
im 
2^54 
•19 
•46 
Mean A 
Interpolation 
6-5 
16^2 
1-07 
M8 
•05 
•08 
Now these results show that for the death-rate and the still-born rate the parabolas give 
better results for extrapolation than the straight lines, but that both representations have a 
mean error much in excess of the average mean error of interpolation within the range on 
which the calculation is based. An examination of the diagram shows that both rate curves are 
actually inadmissible a little beyond the range, for we see that either calculated rate tends to 
fall beyond the beginning of the range and rise after the end of it ; there can hardly be a doubt 
that the very reverse of this must represent the actual state of affairs. In the case of the 
population curves this divergence from the facts does not occur; the total population falls 
before the beginning of the range and rises after the end of it as we should anticipate. But we 
see that the great closeness of the calculated to the observed population within the range has 
been gained by immensely emphasising the fall before and the rise after the range, so that the 
curve becomes worthless for purposes of extrapolation. Indeed while the parabola within the 
range is two to three times as good as the best representative line, the latter is indefinitely 
better than the former for extrapolation, giving indeed extremely good results. We are thus 
forced to the conclusions that : 
(i) Empirical formulae which fit extremely well within the range of observation may give 
very bad extrapolation results only just outside that range. 
(ii) Of two curves that which gives by far the better interpolation results, may give by far 
the worse extrapolation i-esults. We cannot argue from excellency of fit within a given range to 
a fitness for prophesying what occurs even just outside that range. 
Generally we belie\ e that while empirical formulae — including in that term even general laws 
like Hooke's or Boyle's — may be excellent for interpolation within the range of actual observation, 
they cannot be used in either physics or vital statistics without extreme caution, if indeed they 
can be used at all, to predict what will occur even just outside that range. While inter- 
polation and graduation can be satisfactorily carried out by the use of a variety of empirical 
curves, there is no corresponding method of extrapolation unless we have some solid reasons 
for assuming that the phenomena can only be represented by a formula or curve of a very 
definite tyjJe. 
