Calculating like a 19th-Century actuary

As the bicentenary year of Benjamin Gompertz's Law draws to a close (Gompertz 1825) it is salutory to recall what calculating involved for Gompertz and his contemporaries. Not much had changed since logarithms had been invented, two centuries before, and arithmometers were still some decades in the future (Richards 2025). Logarithms it had to be. I think I am of the last generation to have been taught the use of logarithms at school, with the realistic prospect (at the time) of actually using them, so as a service to readers who might find themselves without a calculator to hand, here is how it was done. The example below is, appropriately, to calculate a hazard rate (or force of mortality as Gompertz called it) at a single age.

You need logarithm tables, giving values of \(\log_{10}(x)\) for \(1 \le x \le 10\). Also a quill pen and, should it get dark, a candle.

Exercise

Given that \(\log_{10}(3.8) = 0.5798\), calculate:

\[\mu_{40} = 0.0005 + \underbrace{0.000075858 \times 1.09144^{40}}_{\mbox{Gompertz Term}}\]

Solution

(Table lookups are indicated)

• \(\log_{10}(1.09144) = 0.038 \qquad \color{blue}{\bf (Tables)}\)
• \(\Rightarrow \log_{10}(\log_{10}(1.09144)) = \log_{10}(3.8) - 2 = -1.4202\)
• \(\log_{10}(40) = 1 + \log_{10}(4) = 1.6021 \qquad \color{blue}{\bf (Tables)}\)
• \(\Rightarrow \log_{10}(40) + \log_{10}(\log_{10}(1.09144)) = 0.1819\)
• \(\Rightarrow 10^{\log_{10}(40) + \log_{10}(\log_{10}(1.09144))} = 1.5202 \qquad \color{blue}{\bf (Tables)}\)
• \(\log_{10}(0.000075858) = \log_{10}(7.5858) - 5 = -4.12 \qquad \color{blue}{\bf (Tables)}\)
• \(\Rightarrow \log_{10}(\mbox{Gompertz Term}) = -2.5998 = 0.4002 - 3\)
• \(\Rightarrow 10^{\log_{10}(\mbox{Gompertz Term})} = 0.002512 \qquad \color{blue}{\bf (Tables)}\)
• \(\Rightarrow \mbox{Answer} = 0.003012.\)

Note that the third and fifth table lookups are in reverse, you have \(\log_{10}(y)\) and you want to back out \(y\), which calls for a  spot of linear interpolation (in your head or otherwise).

All that effort to produce a Makeham hazard function at just one age. And I used four-figure log tables, many fewer than would actually have been used. Of course, things could be speeded up a lot by having some tables of pre-calculated values to hand, depending on the problem that required the calculation in the first place. I am sure you could turn the above into an algorithm to compute a whole life table's worth of values of \(\mu_x\). Then the real task might have been, say, to value a widow's fund for a few hundred Scottish ministers (Dunlop 1992) or a few dozen German professors (Dunnington 2004). No wonder the Insurance Companies Act 1870 in the UK mandated valuations only every five years for new companies, or ten years for older companies.

Perhaps we children of the computer age have got just a little bit spoiled?