particular case may afford, I think, a certain presumption for the long run. Suppose the sign of c' to be given, e.g. +, the law of decreasing returns prevailing. Then in the long run of cases-while e' is now positive, now negative in sign, now large, now small in absolute quantity for a majority of those cases an increase of e would be attended with an increase in the denominator of the above-written

expression for P, or rather what it becomes when both numerator

Δτ

and denominator are divided by e, and, therefore, a decrease of the ratio under consideration. Conversely, when the law of increasing returns prevails, c' being negative, an increase of elasticity is likely to be attended with an increase in the efficacy of taxation to raise price.

To the extent of the former clause I have to retract my original statement. But I am still able to affirm universally, without reference to the law of cost, the contradictory of Professor Seligman's theory that "the greater the elasticity of demand the more favourable-other things being equal-will be the position of the consumer"; if the situation of the consumer is tested, as it ought to be, not so much by the rise of price as by the loss of consumer's surplus. Employing the proper criterion of the consumer's welfare, we may affirm, "the greater the elasticity of demand the more unfavourable-other things being equal-will be the situation of the consumer." "1

For the purpose of obtaining propositions in Probabilities, as the preceding may be described, I submit that symbols seem to have an advantage even over diagrams-not to speak of numerical illustrations. Thus it may be objected to our diagrammatic proof 2 of propositions (2) that some a priori knowledge derived from analysis is required to guarantee the legitimacy of our supposition that the law of cost only is varied, other things being preserved constant. A diagrammatic proof of proposition (3) would be even more precarious.

(3) One more of Professor Seligman's general reflexions :

1 The loss of consumers' surplus consequent on raising the price from p to p+Ap is (Cf. Giornale degli Economisti, 1897, p. 314) x▲p; where x is the amount produced, which, by Cournot's equation (3) of ch. V, = e( p −c). Therefore the loss of

2

€2

1

This expression for the loss of consumers' surplus is always positive, both the numerator (= AT÷e) and the denominator (for the reason given in note 1 to p. 312) being positive. The loss is in the long run greater the greater e is, since one term of the denominator becomes less as e becomes greater; while other term of the denominator which involves e may be treated as inoperative, on an average, in the long run of all possible values (positive and negative) of e', in our ignorance of e'. Thus the loss of consumers' surplus is likely to be greater the greater the elasticity.

2 Above, p. 297.

e

"It may even be doubted whether the mathematical method has independently discovered any important principle susceptible of practical application that could not have been also expressed in every-day language."

Those who have followed the preceding discussion may be disposed to admit that, if the mathematical method does not itself discover important practical principles, it may at least be usefully employed to test the principles which a distinguished practical economist regards as important. If it is worth his while to employ some pages of economic analysis and numerical examples in endeavouring to prove those principles, it is worth our while to employ some lines of symbol in endeavouring to disprove them. Thus it is argued by Professor Seligman:

"If a high tax, for instance, be imposed on the passenger tickets of a railway, subject to the law of increasing returns where the most profitable business happens to be the passenger traffic, and where an increase of fares would mean a considerable falling off in travel, the resulting abandonment of several passenger trains would mean a considerable increase of the percentage of fixed to operating expenses, and therefore a great fall of profits. The railway will therefore add as little as possible of the tax to the fare."

If this thesis deserves attention, so does the anti-thesis, that if the law of increasing returns holds good1 and the demand is elastic,2 a tax on railway tickets is likely to be particularly hurtful to passengers. In these days when the system of monopoly is, so to speak, on its trial, a certain practical interest attaches to Professor Seligman's theory that, whereas

...

"the condition most favourable to a monopoly is that of decreasing cost or increasing returns" "the tendency is . . . that less of the tax will be shifted to the consumer than under any other proportions in the ratio of product to cost" (p. 210).

This is, indeed, a "comforting doctrine to the consumer" (Ibid). The doctrine is certainly important, if it is true. If it is not true, though supported by such high authority, is not the contradiction of it important?

These negations are also affirmations, but not very confident ones. It is as if an opponent should prophesy that the last week of April or May would be the coldest part of the month. The reply is that what we know about the matter points in contrary direction: there is a constant cause making for greater heat-namely, the position of the earth relatively to the sun-in the latter part of each month; though doubtless that tendency may be counteracted by unpredictable vicissitudes of weather. What if the more abstract part of political economy, like the more sublime part of astronomy-that which contemplates the mechanism of the heavenly bodies external to our system-were not at present susceptible of direct practical application, the mathematical 1 Even in the author's sense. Above, p. 302. 2 Above, p. 313.

theory of economics might still confer a benefit analogous to that which the mathematical theory of astronomy conferred when it discredited the pernicious pretensions of the astrologers. There are those who think that even of the received economic analysis the most important function is negative. Thus Mr. Leslie Stephen :

"Political economy, as I venture to think, has been especially valuable in what I have called its negative aspect. It has been more efficient in dispersing sophistries than in constructing permanent theories. Economic writers have exploded many absurd systems. They have so far cleared the way for an application of sounder methods. But the complexity of the problem is so great. . . ."1

The sort of sophistry which has been eradicated from the general field of economics by the received organon finds a still virgin soil in the nooks and corners of which the cultivation requires the implements of mathematics.

The trenchancy of this criticism is not inconsistent with the diffidence which is proper to an inexact science, and the respect which is due to a high authority. For on the one hand, the region of hypothetically abstract theory, to which this polemic is confined, forms the one territory of economics in which issues may be fought out without compromise, there being a right diametrically opposed to the wrong. And on the other hand, it is no discredit to the ablest combatant, when he is unprovided with the proper weapons, to succumb. F. Y. EDGEWORTH.

MORTALITY IN EXTREME OLD AGE

IN times past longevity was a favourite subject among authors on vital statistics. Curious lists with particulars as to very old persons were drawn up and discussed, and the aspects of longevity under various circumstances traced. Thus the well-known German statistician, Süssmilch, author of the voluminous work Die Göttliche Ordnung, deal at length with this question (1.c. §§ 481 seq.). The famous naturalist A. v. Haller, likewise, in his Elementa Physiologia Corporis Humani, deals with longevity generally as well as with "longævitas hominis " before and after the flood, pointing out several causes, as heredity, frugality, or contemplative habits of life. Later on C. W. Hufeland wrote his renowned Makrobiotik (1796), William Barton his Observations on the Probabilities of the Duration of Human Life (1793), and James Easton his Health and Longevity (1799).

A hundred years ago there was thus a series of observations on longevity at hand, which might have led to interesting conclusions, if they had only been correct. Unfortunately this was not the case; many of the most famous records of longevity have been reduced into absurdity, or they have at least lost much of their romantic flavour. 1 Life of Fawcett, p. 149.

Some curious instances will be found in a paper by an American author, J. E. Worcester (Remarks on Longevity, &c., 1833): "In a magazine published at Philadelphia in 1804, it was stated that Samuel Bartrow died at Boothbay, Maine, at the age of 135. But instead of this it appears that a man of the name of Barter died at that place at the age of 105. Several newspapers and journals, in 1823, mentioned the death of a Moor, of the name of Yarrow, at Georgetown, Columbia, at the age of 135, but it has been found that his age was only about 85."

Undoubtedly several other records of centenarians are tolerably accurate. If we confine ourselves to the comparatively youthful persons of 100-105 years, we shall find many reliable cases, as that of Sir Moses Montefiore, or of the French naturalist Chevreuil, and even if we go to much higher ages, there is often good evidence, as in the case of the Norwegian Drakenberg (1626-1772), who married when 111 years old, and as a widower of 130 proposed to marry again, though without success.1 But these records are neutralised by other more or less doubtful cases, and at the end of the 19th century statistics of longevity are still in a very incomplete condition.

No wonder then that life tables generally take very small notice of extreme old ages. Sometimes they stop abruptly, for instance at 100 years of age, or the rates of mortality in the highest ages of life. are not based on observations, but on some fiction, generally that mortality is increasing with age according to a simple mathematical function till all have died out. But probably none of the authors of life tables would look upon this scale of probabilities otherwise than as a conventional fashion of bringing a tale to an end, nor is it of any practical consequence in life assurance or similar matters what are the rates of mortality among very aged persons, the number surviving being so small that it is of nearly no influence on the premiums charged by the life offices what is the law of mortality in this period of life. No wonder therefore that modern statisticians have taken so small notice of this question.

But even if life tables have no great value for business purposes, they may have an intrinsic value, and I hope the following investigation will be found of some interest for students of vital statistics.

If we consult the unadjusted life tables we sometimes find a curious interruption in the general law of mortality. The following table shows the probabilities of dying in the course of a year for males of various ages according to the German and Norwegian life tables 1871-1881.

It will be seen on inspection of these numbers, that the chances of dying within the next year are not increasing according to the German unadjusted table and, from 95 years of age, rather decreasing according to the Norwegian table. The question arises whether this must be ascribed to the defects of the numerical observations or whether there 1 Dansk Biografisk Lexikon, IV, 1890, pp. 327-8.

is a real influence counteracting the general tendency to an increase in mortality according to age.

It is evident that a person like Drakenberg could hardly have existed if 50 per cent. or more of the old persons alive died every year. Very few persons in a million will reach 100 years of age, but if out of these one half died yearly, only 1 of the centenarians would reach 110 years, and one out of a million 120. If there are some rare cases of persons with an extraordinary age, the law of mortality must have a form which renders it possible to meet these exceptions every now and then.

But, on the other hand, we cannot rely on the official statistical data without any revision. For instance, if only a few persons were erroneously made 20 years older than they are, we might observe a conspicuous influence on the law of mortality in the very old ages. According to the Norwegian table there will be 33,695 males alive at 70, 15,347 at 80, 2,440 at 90, and 70 at 100. Let us now suppose that only 1 per mille of males, 70 years old, was erroneously registered as 90 years old, the number of males of this age would be 2474 instead of 2440; the error would thus be of little consequence. Ten years later there would be about 15 persons alive out of the 34 persons whose age was wrongly registered, and if the mistake is still undiscovered, the number of centenarians will be increased from 70 to 85. The influence of this apparently insignificant error will thus be constantly greater every year. The probability of dying before a year for a person of 95 years would be nearly unaltered, but for a centenarian it would sink from 41 to 36 per cent., and 5 years later it would be about 30 per cent. instead of 60 per cent. It will thus be necessary to be very careful in order to draw correct conclusions out of such observations.

The Norwegian Central Office of Statistics adopted the good plan