Reading the (actuarial) runes

While researching my paper on the actuarial origins of survival models (Richards, 2026), I had cause to re-read Gompertz (1825).  Strictly speaking I should say "re-try", as my first attempt to read his paper as an undergraduate failed.  Part of the reason for this was notation that was incomprehensible to me in the late 1980s.  Figure 1 shows an example:

Figure 1. Expression from Gompertz (1825, p.539).

For the modern reader, Figure 1 bears more than a passing resemblance to ancient hieroglyphics. Here is an actuarial Rosetta Stone:

• \(\lambda\) represents the logarithm to base 10.

• \(\mathscr{C}\) is a script letter C (I think) that represents the item to be solved for.  Modern mathematicians would nowadays probably just label it \(y\).

• \(b\) is the current age, which today's actuaries typically denote by \(x\).

• The squiggle represents a level life annuity paid contingent on the survival of the life aged \(b\).

• The \(1,05^{-1}\) above the squiggle represents a discounting factor at 5% per annum. Modern actuaries denotes this by \(v\).  Note that Gompertz used the comma as a decimal separator, a convention still in use in continental Europe today.

• The first number below the squiggle refers to the payment (1 for £1), with annual payment implied.  The lowest number below the squiggle signals when the annuity payment starts: 0 for immediately, 1 for commencement in a year's time.

In modern actuarial notation, Figure 1 simply says:

\[\log_{10} C = \log_{10} a_b^{\rm 5\%} + \log_{10} 1.05 - \log_{10} \ddot{a}_b^{\rm 5\%}\]

where \(a_b^{\rm 5\%}\) denotes an annuity payable annually in arrears to a life aged \(b\), valued at 5% interest, with \(\ddot{a}_b^{\rm 5\%}\) the equivalent annuity paid in advance.

There is another trap for modern readers, as exemplified by this equation in Gompertz (1825, p.527):

\[\lambda(,07985\}-\lambda(2,41243) = \bar{2}.519874.\qquad(1)\]

Those currently aged below 60 might assume that the bar above the 2 in equation (1) is just a curious place to put a negative sign, i.e. that the right-hand side is \(-2.519874\). This is baffling because:

\[\log_{10} 0.07985 - \log_{10} 2.41243 = -1.48018.\]

The explanation is that \(\bar{2}.519874\) is bar notation for \(0.519874-2\).  Gompertz is using tables of logarithms as follows:

\[\begin{align}\lambda(,07985)-\lambda(2,41243) &= \log_{10}(0.07985)-\log_{10}(2.41243)\\ &= \log_{10} (7.985\times 10^{-2}) - \log_{10} (2.41243)\\ &=\log_{10} (7.985) - 2 - \log_{10} (2.4124)\\ &= 0.90227 - 2 - 0.38245\\ &= 0.51982 -2\\&= -1.48018 {\rm \ or\ } \bar{2}.51982\end{align}.\]

The final answer above agrees the first few significant digits of the right-hand side of equation (1), but it doesn't agree exactly due to the limitations of Gompertz's logarithm tables.

Thanks to Prof. Angus S. Macdonald for explaining the mechanics of log tables to this bemused correspondent.  As Angus wrote in an earlier blog, calculating like a 19th century actuary was not easy.  Reading what 19th century actuaries wrote is no picnic either.