Comparison of Curves representing Rates of Sickness.

A principal motive for laying down a curve to represent the mean annual rate of sickness was to afford an easy method of comparing it with the results of observations, made some years since, on the rate of sickness among Friendly Societies in Scotland; which results were published by the Highland Society in one of their Reports, and they are also given in the Report of the Select Committee of the House of Commons on the subject of Friendly Societies, ordered to be printed on the 5th of July, 1825.

The following is the table alluded to, as deduced from the experience of the Societies in Scotland.

The line, marked in the foregoing plate, coincides with the last-mentioned table, and the close agreement as to the relative rate of sickness, at all the ages embraced in the term compared, is a fact strongly tending to establish the credit of both Tables; although the actual quantity of sickness shown in the Scotch Table is very considerably less than the actual quantity experienced among the English Societies; which circumstance would necessarily render computations based on the former quite inapplicable to the purposes of Societies in England.

The following statement shows the ratio of the mean annual sickness in the English Societies to the sickness in the Scotch Societies, for the several ages before compared.

In the sixth edition of Dr. Price's Observations on Reversionary Payments, edited by Mr. W. Morgan, Actuary to the Equitable Assurance Society, page 473, it is stated that certain tables there given were computed by Dr. Price at the request of a Committee of the House of Commons; and further, that it was Dr. Price's intention to have published them had he lived to complete that edition of his work. Mr. Morgan adds, "In order, therefore, to fulfil his intentions, as well as to preserve those valuable fruits of his labours from being lost, I have inserted them, together with his own explanations of their use and construction. The first table shows the weekly allowances during incapacities of labour produced by sickness or accidents, and the correspondent weekly contributions necessary to entitle persons to those allowances."*

Then follows the "Supposition on which the Table is formed." First, that in societies consisting of persons under 32 years

* Since the above paragraph was written, death has terminated Mr. Morgan's longcontinued and valuable labours as a man of science and of extensive practical knowledge.

of age, a 48th part of them will be always in a state of incapacitation by illness and accidents; and therefore entitled to allowances proportioned to their contributions. Various reasons, and particularly the experience of Friendly Clubs, determine me to believe, that the proportion of the sick to the well in such a society will not be so great as this, and, consequently, that a weekly allowance during sickness will be more than supported by weekly contributions not exceeding a 48th part of that allowance."

66 Secondly, It is supposed, that from the age of 32 to 42 this proportion increases to one quarter more than a 48th part; from 43 to 51 to one half more; from 52 to 58 to three quarters more; and from 59 to 64 to double. The reason of assuming this rate of increase is, that the probability of the duration of human life decreases after 30 nearly in this manner, or so that a person of the age of 60 has but half the probability of living any given time that a person at 32 has, and consequently must be then doubly subject to the causes that produce sickness and mortality."

It is not very obvious in what sense the term "probability of living any given time" is intended to be understood. It does not agree with the term as generally used, and as it is used in page 479 of the same volume of Dr. Price's own work; inasmuch as the probability of living one year for instance, according to the Northampton Mortality, (the Table used by Dr. Price,) at age 32, is 98229, and at age 60 it is ·95976, while the quantity of sickness experienced at the latter age is distinctly said to be twice that experienced at the former age. This increase in the sickness is very nearly in the inverse ratio of what is called the expectation, or mean duration of life, there is therefore sufficient reason to think that such was the term intended to be adopted; and that after age 32 the quantity of sickness was thought to increase with the age, inversely as the expectation of life. The expectation at age 32 is 27.24 years, and at age 60 it is 13.21 years. On this last supposition the table in the next page, showing the quantity of sickness experienced at all ages from 20 to 70, was computed; and the curve represented by the broken line in the last plate, is made to correspond therewith. It will immediately be seen how widely the assumption of Dr. Price differs from the facts as shown from actual observations; and the circumstance tends to show how little reliance is, in similar cases, to be placed on hypotheses, however ingenious those hypotheses may be.

Mean Annual Sickness as deduced from the hypothesis of Dr. Price.

In all the subsequent calculations which relate to allowances during sickness, the Table at page 70 has been used, from a belief that at present no similar data exist of equal authority; but it by no means follows, that our exertions should be lessened to add other observations to those already collected. Much remains to be done, particularly the obtaining correct data from which to determine the rate of sickness which prevails from age 70 to the end of life; and also the difference in the rate of sickness which takes place among the inhabitants of towns and persons living in agricultural districts, as well as among persons of unlike occupations, supposing such differences to exist: for which purposes the returns obtained are not sufficiently extensive.

COMBINATION OF THE PRINCIPLES OF
COMPUTATION AND DATA.

We will now proceed to show how the doctrine of interest of money and the doctrine of probability, as applied to the duration of life or the occurrence of death or of sickness, are combined in ascertaining the contributions which ought to be paid to provide for an annuity in old age, for a payment at death, and for an allowance during sickness.

Value of a Life Annuity.

It was demonstrated at page 36, that the money value of the expectation of an individual, as respects the happening of a particular event, is equal to the product of the probability that such