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Mortality & Life Insurance

The creation of the first mortality tables permitted annuities to be put on a firmer foundation. Thus begins works devoted to the computation of such. We include here selections from  Aubrey's Brief Lives to provide glimpses of some personages. These lives are published as Volume I (A-H) and Volume II (I-Y) under the editorship of Andrew Clark in 1898.

Domitius Ulpianus

Ulpian was a celebrated Roman jurist. He constructed a table of the remaining years of life to be assigned to an individual of a given age. The purpose of this rule was to assist in the evaluation of estates or for the duration of a usufruct in the matter of succession of inheritances.

Dio Cassius in writing of the events of 40 BC (Roman History Book XLVIII.33.5) says that the lex Falcidia was enacted by Publius Falcidius while he was Tribune of the Plebes. Its terms are that if an heir feels burdened in any way he may secure a fourth of the property bequeathed him by surrendering the remainder. Prior to the l. Facidia heirs to an insolvent estate were liable for all debts without limit. The essential feature of this law was that legacies or bequests could not exceed three-quarters of the total estate. Equivalently, a testator could not deprive his legal heir of more than three- fourths of the estate. Legacies in excess of three-fourths were scaled down pro rata.

The l. Facidia was applied separately to each heir. A legatee may be required to give security for return of what might be paid in excess of  what was due under the l. Facidia or for return of the legacy if the supposed heir was evicted. Security was also given so that the full value of a usufruct may be returned at the expiration of the usufruct.

This law remained in force into the sixth century since it was incorporated in the Institutes of Justinian.

John Graunt & William Petty

The Observations on the Bills of Mortality, written by John Graunt, was published in 1662. A facsimile of the first edition may be found at the website of Ed Stephan.

William Petty was a close friend of Graunt (See Aubrey's Brief Life) who wrote on any number of political topics. He coined the term "political arithmetic." His essay Political arithmetic was likely begun in 1671 but its completion not earlier than 1676. This work concerns the extant and value of lands, people, buildings of Great Britain and how this relates to neighbors of Holland, Zealand and France. It was printed in 1690. Political arithemetic as a discipline was developed by Petty, Gregory King (1648-1712) and Charles Davenant (1656-1714) in England, Vauban (Sébastien Le Prestre, Seigneur de Vauban, 1633-1707) nick-named the French "Petty," Nicolas Struyck (1687-1769) in the Netherlands, and Johann Peter Süssmilch (1707-1767) in Germany.

There was, in fact, debate as to authorship with some asserting that it was the product of Petty rather than Graunt. For this reason, it is natural that it be Included among his collected works. Readily available is a reprinting of the fifth edition of the Observations. For this see the Economic Writings of Sir William Petty edited by Charles Henry Hull (1899). In Volume I may be found the material on the life of Graunt. Volume II contains the Observations.

The brothers Huygens, Christiaan and Ludwig, engaged in correspondence regarding life expectancy derived from Graunt's table.

  1. Greenwood, Major, "Graunt and Petty," Journal of the Royal Statistical Society, Vol. 91, Issue 1 (1928), pp. 79-85.
  2. Greenwood, Major, "Graunt and Petty--A Re-statement," Journal of the Royal Statistical Society, Vol. 96, Issue 1 (1933), pp. 76-81.
  3. Willcox, Walter, "The Founder of Statistics," Revue de l'Institut International de Statistique, Vol. 5, No. 4 (1938), pp. 321-328.


Johann Van Hudde & Johan (Jan) de Witt

In the Histoire des Mathématiques originally written by Jean-Étienne Montucla and later published with additions by J. de la Lalande (See Volume 3, Part V, Book I, page 407 (1802)) it is mentioned that Johann Hudde, Burgomeister of Amsterdam, (1628-1704) had written on annuities but the title was unknown. The work however mentions Waardye van Lyf-renten naer proportie van  Losrenten or The Value of life-annuities in proportion to redeemable annuities by Jan de Witt (The Hague, 1671).

De Witt's treatise was essentially lost until recovered by Frederick Hendricks among the papers in the state archives of Holland around 1850. He published a translation of it in The Assurance Magazine and Journal of the Institute of Actuaries as part of a larger study entitled "Contributions to the History of Insurance and the Theory of Life Contingencies."  These appear in Vol.2 (1852) pp. 121-150, 222-258, and Vol. 3 (1853), pp. 93-120. The second section contains the treatise of de Witt and the third section, containing the correspondence of de Witt with Hudde as recovered by Hendricks, is the source of our knowledge of the role played by him.
The Treatise on Life Annuities  and a biography of de Witt may be found at the York site. See also A Sketch of the Life and Times of John de Witt by Robert G. Barnwell (1856) which also reprints the translation by Hendricks.

 Leibniz remarks when speaking of the study of games of chance:

"Mr. le Pensionnaire de Wit a pousse cela encore davantage, & applique à d'autres usages plus plus considérables par rapport aux rentes de vie: & Mr. Huygens m'a dit, que Mr. Hudde a encore eu d'excellentes méditationes là-dessus, & que c'est dommage qu'il les ait supprimées avec tant d'autres. Ainsi les jeux mêmes mériteroient d'étre examinés: & si quelque Mathématicien pénétrant méditoit là-dessus, il y trouveroit beaucoup d'importantes considérations; car les hommes n'ont jamai montré plus d'esprit que lorsqu'ils ont badiné."

That is,

"The Pensionner de Witt has pushed this yet further, & applied to some other uses more considerable with respect to annuities: & Mr. Huygnes has said to me, that Mr. Hudde has yet had excellent thoughts on that subject, & that it is a pity that he has suppressed them with so many others. Thus the same games would merit to be examined: & if some shrewd Mathematician would meditate on that subject, he would find many important considerations; for me have never shown more spirit than when they have play."

In addition, among the correspondence of Jakob Bernoulli and Leibniz we find the futile attempt by Leibniz to obtain a copy of the treatise of de Wit.

Edmund Halley

The life table of Graunt was certainly unsatisfactory and many saw the need for one constructed from real data. Halley, (Aubrey's Life of Halley) for whom the comet is named, made use of the Breslau (Wroclaw, Poland) tables in the important memoir "An estimate of the Degrees of the Mortality of Mankind, drawn from the curious Tables of the Births and Funerals at the City of Breslaw; with an Attempt to ascertain the Price of Annuities on Lives."  This is followed by "Some further Considerations on the Breslaw Bills of Mortality." Both papers were published in the Philosophical Transactions of the Royal Society Vol. XVII (1693) pp. 596-610 and 654-656 respectively. The links are to the University of York site.

One observation should be noted here and that is that some debate arose over the initial population. Graunt speaks of 100 quick conceptions. Halley begins with a population of 1000, but it is not clear if this is the number of births or the number who arrive to age 1 year.

Further Developments

We mention in passing that Abraham de Moivre and Thomas Simpson after him wrote on Annuities on Lives.

Finally, there is Johann Peter Süssmilch (1707-1767) whose Die Göttliche Ordnung was published in 1741, its full title being Die Göttliche Ordnung  in den Veränderungen des menschlichen Geschlechts, aus der Geburt, dem Tode und der Fortpflanzung desselben erwiesen or The divine order proved in the changes of the human sex, from the birth, death and the reproduction of the same.  A second edition of two volumes expanded from an original of 365 pages to 576 and 625 pages respectively and exclusive of tables appeared in 1761-2.  A third edition from the author appeared in 1765. The edition for which links appear below is that revised by Süssmilch's son-in-law and edited by Ch. J. Baumann in 3 volumes of 1775. 

A study of Süssmilch's statistical work by Frederick S. Crum, "The Statistical Work of Sussmilch," appeared in Publications of the American Statistical Association, Volume VII,  No. 55, pp 1-46. On page 42, the preface to the first edition by Süssmilch is quoted in part and that is reproduced here:

"Because these remarks were new to me, a desire was awakened for further investigation, and as the great utility of these truths was plainly evident, I became attentive to everything which might strengthen them. When I came back to Berlin from the University there fell into my hands some additional lists, both from Berlin and for the whole country. To my great satisfaction I observed an almost complete agreement of our countries with England in these matters. I got together everything that I could find. The contributions which had been made for some years in Breslau to medicine, physics, and other sciences contained many lists ... which in great measure confirmed the order observed by the Englishmen... As I was gradually led ever farther, the writings of the Lord-Mayor Graunt and of the Knight Petty fell into my hands... To Graunt belongs the highest praise, for he first broke the ice over these new truths, and he first tried, in the search for them, to make use of the London lists. Petty soon followed, and in his attempts in political arithmetic not only accepted and confirmed many of Graunt's propositions but also clearly demonstrated their utility in politics and administration."

Die Göttliche Ordnung
Volume I,  Volume II, and Volume III of 1775-6. The text of 1741 has been rendered into French by Jean-Marc Rohrbasser and published as L'Ordre Divin by L'Institute National d'Études Démographiques (INED) in 1998.