Again, the value of the first part of the correction of the annuity is

The sum of (16) and (17) gives the total correction of the annuity,

viz.,

Adding this correction (18) to the annuity (15) we get finally for the present value of the complete reversionary annuity payable yearly, the following expressions, viz.,

Here, by substitution of i=e-1, and expansion, the last term is

360 (1+

21

the value of which is practically quite insensible, and may therefore be rejected.

By other approximate substitutions which in practice may be considered as exact, the formulæ for the complete reversionary annuity (19) become ultimately

which last agrees in form with the general expression determined, by a most simple and comprehensive process, in my paper on " An Improved Theory of Annuities and Assurances," where (Journal, vol. xv, page 113) it is shown, generally, that the value of the complete reversionary annuity of practice, when payable in m instalments in each year, is

This last formula is derived by a general method, without any supposition of equal decrements in each year of life, or indeed any supposition whatever affecting the law of mortality. The perfect identity in the form of the resulting expression, when the annuity is payable yearly, must not be understood to imply an agreement as to absolute value. The values of the continuous quantities a, ary, Al, &c., which enter the formulæ (20), are severally affected by the imperfection of the hypothesis of equal decrements, and through them the corresponding small error of the annuity is imparted to the numerical result. The quantities which exclusively depend upon yearly intervals, viz., ɑx, ɑxy, Ar, &c., are, however, independent of the hypothesis, but the formula for the annuity, expressed in terms of them, involves the value of the conjoint assurance and differs from the true formula in form as well as in value.

xy

It is my intention shortly to draw up for publication a treatise, embracing a full development of this important theory, and its practical applications.

The errors due to the hypothesis of equal decrements are readily found from what precedes. For example, to find the errors of A and a, we have (Improved Theory, formula (29)),

б

According to the hypothesis in question, the value is Az. Therefore the error in the value of the continuous assurance Ã, is

And since ā,= ¦ (1—Ãx), the error in the value of the con

1

tinuous annuity a is (ux-iAx). The errors of the joint and

12

xy

survivorship assurances Ary, A, since they depend on the relations (7) and (8), will involve the value of the conjoint assurance Way.

From the reductions that have been so far accomplished, we conclude, generally, that the assumption of equal decrements in each year of age, the hypothesis usually adopted, for convenience or expediency, in treatises on annuities and assurances, although very interesting and suggestive as a subject of discussion, does not yield us any additional facility of calculation as a set off against the small errors and imperfections that are necessarily engendered in so many of the quantities that come under our consideration.

On Mathematical Statistics and its application to Political Economy and Insurance. By Dr. THEODOR WITTSTEIN, Actuary of the Hanover Life Insurance Company. Translated by T. B. SPRAGUE, M.A.

PART I.
INTRODUCTION.

Concerning a New Science, being a Paper read before the MathematicoPhysical Section of the 40th Natural Philosophy Congress, held at Hanover in 1865.

It is well known that the number of the physical sciences is continually increasing, by a process of development from within. We constantly see divisions and branches of existing sciences

cultivated and developed until they acquire a separate existence as new and independent sciences. For it has long been impossible for a single person to master the whole, or even a considerable part of these sciences, and the necessity for a division of labour is continually becoming more and more felt here, as elsewhere. It is, however, comparatively seldom that the number of the sciences is increased by the addition to them of an entirely new one, and therefore the case of which I am about to speak appears to deserve our particular attention. For I shall presently prove that a department of science, which has not hitherto belonged to the physical sciences, is now being so developed and perfected, that in a very short time it will claim admission into the number on equal terms with the others. This department of science is Statistics. The science of Statistics in its present form has now existed a full century. It owed its birth, as well as its name, to Achenwall of Göttingen, about the middle of the last century. It may be defined as the compilation and comparison of all the noteworthy facts that a State—or, in more general terms, a Society-presents to our view at a given epoch; of what Schlözer designates by a single word "Staatsmerkwürdigkeiten." But to this definition the subsidiary idea soon attached, that the information in question must be given principally in numbers; and thus we see that the present text books on statistics really consist, for the most part, of a collection of tables giving all the information that can be expressed in figures, as to the population, manufactures, agriculture, trade, &c., &c., of a country. In this form, statistics is one of the social sciences, and is especially considered as an important auxiliary to the science of political economy. Now we must bear in mind that, philosophically considered, a collection of tables containing observed facts can make no claim to be considered a science in the proper sense of the word. Such tables only furnish the rough materials from which the science is to be constructed. In scientific phrase, they are only a series of observations, this very word expressing what is the next step required. The problem before us is the same as in all other branches of natural philosophy-to ascend from observations to the discovery of natural laws. In this direction, statistics, faithful

to its definition, has hitherto done scarcely anything. It has, in fact, only drawn the obvious conclusions which result from a first glance at the tables, and, therefore, when it attempts to solve the above problem, it becomes essentially a new science. It has to begin at the point where statistics, as hitherto understood, ceases; and since the new science deals principally with figures, we must

employ mathematics for the solution of the problem. The new science may therefore be called Mathematical Statistics, unless mathematicians should prefer to call it Analytical Statistics, according to the analogy of Analytical Optics, Analytical Mechanics, &c. This will be made still clearer by a comparison with another instance of which the circumstances are more generally known. The astronomer makes observations and collects them in the form of tables; but if he regarded his work as then completed, astronomy could never have claimed to be called a science. It would have remained simply in the stage occupied by statistics as hitherto understood. Such nearly was the position of astronomy when Tycho Brahe made his observations on the planet Mars, which have since become so celebrated. Then came Kepler, who from these observations deduced his well known laws. In Kästner's happy phrase, "Tycho quarried the marble, and Kepler chiseled the statue.' It was this step, afterwards completed by Newton's theoretical investigations, that first raised astronomy to the rank of a science, properly so called; and this step has not yet been made by statistics. It must not, however, be supposed that a Kepler only is required to raise statistics to the dignity of a science. Its Tycho has not yet appeared; for statistical facts and figures are as yet, with scarcely an exception, of very doubtful value, and (as I shall presently show) unsuited for mathematical investigation. Nay, keeping to the same metaphor, it may be said that its Copernicus is still to come, who shall for the first time give us a general view of the facts we ought to observe. Statistics is now in its infancy-at the same stage as astronomy was when it was nothing but astrology and drew horoscopes. It still happens every day, that people prove by statistical figures whatever they choose, the only thing necessary for the purpose being a kind of dexterity, happily expressed in the French phrase, "grouper les nombres."* This will be amended as soon as mathematics, with its inexorable logic, shall have mastered the statistical materials. The calculus of probabilities is the special branch that must here be applied, and that not in the sense so commonly attached to the phrase by statisticians, according to which it is a calculus that gives inexact results as distinguished from exact calculations, but that ingenious science which is as completely based on logic as any part of

*Say gives one of many instances in his Traité d'économie politique: "The French Minister of the Interior, in his report for 1813, a time of calamity, when commerce was destroyed and the national resources of every kind were rapidly diminishing, boasts of having demonstrated by means of figures that France was in a condition of prosperity greater than it had ever before enjoyed."