Euler, Leonhard

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EULER, LEONHARD

(1707–1783)

Arguably the greatest mathematician of the eighteenth century, Leonhard Euler, a Swiss, made basic contributions to every branch of mathematics then being studied. His enduring place in the history of demographic analysis is assured due to his 1760 formula for a stable population.

An analysis of the formula begins by considering a population with fixed and known death rates at each age and a constant number of births per unit time in all generations. From the death rates, it is possible to create a life table giving the probability of survival to age x, denoted l(x). If the number of births is b(0), the population in the age group from x to x + dx, where is small, is b(0)l(x)dx, a constant over time. If the number of births is increasing at an exponential rate, r, t years into the future the number of births would be b(0)e^rt. Similarly, t years in the past, the number of births would have been b(0)e^-rt. Thus, taking the formula for births t years ago, and multiplying by l(x), the proportion of births surviving to age x, gives a stable population with the age distribution b(0)l(x)dx = b(0)l(x)e^-rx. Euler was the first to develop this formula, which is the starting point for much subsequent population modeling.

If survivorship remains fixed and r(x) varies, then the formula remains the same, but has a different interpretation; and more generally, allows for survivorship to vary as well. Then, given birth and death rates as they vary with time, researchers have the usual formula for population at the end of any time interval. At first this was called forecasting, but the unreliability of the process led to the more modest term projection, meaning simply the working out of the consequences of the assumed regime of birth and death rates.

The theory above stated has been reformulated as a set of partial differential equations more applicable to numerical work. Neither of these continuous models, developed as theory, is convenient for calculation with real data.

While Euler never went beyond fixed rates, he can reasonably be described as the originator of the field of projection and prediction that preoccupies demographers, and even more so, the public at the beginning of the twenty-first century.

Well over a century after Euler, demographer Alfred Lotka (1880–1949), with much wider demographic (and additional environmental) interests, published, in 1907, the renewal equation, the fundamental relationship in population dynamics, that permitted calculation of the rate of increase implied by a regime of birth and death rates.

Euler, called by his contemporaries the Prince of Mathematicians, commanded the respect of kings and the public, and above all of mathematicians. He was a lifelong friend of the brothers Daniel and Nicholas Bernoulli. He held an appointment at the University of Basel. He was invited to Russia by Catherine the Great, and to Berlin by Frederick the Great, both powerful patrons of the arts and sciences during the eighteenth-century Enlightenment. Euler's private life, however, was far from happy. He was blind for almost twenty years before his death. In 1768, he lost many of his unpublished papers when his house was destroyed in a fire. He died of apoplexy.

bibliography

selected works by leonhard euler.

Euler, Leonhard. 1977 (1760). "A General Investigation into the Mortality and Multiplication of the Human Species," trans. Nathan and Beatrice Keyfitz. In Mathematical Demography: Selected Papers, ed. David Smith and Nathan Keyfitz. New York: Springer-Verlag.