The organizing Committee of the Twelfth International Congress of Actuaries which was to have been held in Lucerne in 1940 did me the honour of suggesting that I should give a lecture on the development of insurance mathematics, particularly the mathematics of non-life insurance. Unfortunately the war prevented the Congress, but the individual articles were published in Zurich. Those which relate to the subject mentioned are to be found in Section I which includes papers intended to answer the following questions: ‘What hypotheses underlie actuarial science and to what extent can the applications of the theory of probability and of the theory of risk be justified? Should actuaries use these theories—particularly more recent developments—in practice? Are observations made on insurance data likely to confirm the theories of probability and of risk?’ The articles in this Section are concerned only to a slight degree with non-life insurance and in Prof. Jecklin's summary review he correctly points out that up to the present the mathematician has not generally concerned himself with non-life insurance. He writes: ‘Many problems, in particular those relating to the mathematics of non-life insurance, will be only of academic interest so long as it proves impossible to test the theory by means of appropriate statistical data. We must not forget that interesting and valuable though theory may be—permitting us as it does to study effects in a model form based on specific assumptions—yet for practical purposes (for instance when premiums and reserves are to be calculated mathematically according to the theory of probability and statistics) a mass of recent statistical data must be available.