especially by the Arabs in algebra and in trigonometry. In particular, the work from which our modern term algebra is derived was composed during this period, and the work of several well-known Hindu writers appeared therein. Hence the reader seems justified for having somewhat high expectations as regards the mathematical importance of Gerbert's letter if it actually deserves being called "the first mathematical paper of the Middle Ages." In fact, the non-mathematical reader might be inclined to fear that the mathematical merits of this letter were too great to lie within the limits of his comprehension.

These expectations and fears are apt to be enhanced by the reading of the accounts of Gerbert's letter in some of our most popular histories of mathematics, including the ones already noted. Not only is it stated here that this letter contained a correct explanation for the difference of the results obtained by using two different formulas for the determination of the area of an equilateral triangle, but some of the other statements relating to this letter are sufficiently obscure and misleading to arouse the suspicion that the subject treated therein might possibly be difficult. In various instances the obscurity is increased by the fact that figures of triangles which are not equilateral are given, while the text relates to an equilateral triangle. This is done, for instance, on page 249 of Günther's, Geschichte der Mathematik, 1908, as well as in the three editions of volume I of Cantor's well known Vorlesungen über Geschichte der Mathematik, pages 744, 815 and 866, respectively.

From the preceding remarks the reader will naturally conclude that the present writer does not believe that the letter in question merits to be called "the first mathematical paper of the Middle Ages which deserves this name," notwithstanding the fact that this epithet has been applied to it by eminent authoritities. In fact, the present writer believes not only that the letter does not merit this epithet but that it is of so little mathematical importance as to make it appear ridiculous to make such a claim for it. Moreover, he believes that other statements made about this letter in well-known mathematical histories are strikingly inaccurate. In order to establish the correctness of this point of view, it is necessary to state just what is found in the part of this letter which has been preserved, upon which our view of its merits must be based.

In view of the great claims made for this letter, the reader will naturally be surprised to find that it deals with the very elementary question of finding the area of an equilateral triangle, a question which had been completely solved many centuries before. Gerbert gives here the rule that the altitude of such a triangle can be found by subtracting one-seventh from its side, which is a sufficiently close approximation for many purposes, since the altitude is a/2V3, where a is the

side. He then states, in substances, that the area of an equilateral triangle is one-half of the product of the base and the altitude thus obtained, and he calls this the "geometric rule" for finding such an

area.

Thus far there is nothing surprising in this letter, and no one seems to have claimed much credit for this part, but Gerbert then makes some very inaccurate and foolish remarks about finding the area of such a triangle by another rule, called the "arithmetic rule," and it is just upon these remarks which exhibit a great lack of geometric insight that the high claims of this letter have been based. The inaccuracy of these remarks had been noted by M. Chasles in his well and favorably known Aperçu Historique, 1875, page 506, but notwithstanding this fact various later mathematical authors, including all those noted above, have called them correct in their general histories. In order to appreciate the crudeness of this "arithmetic rule," which is equivalent to the formula 1⁄2a(a+1), it may be noted that in the work of the Egyptian Ahmes, written about 1700 B. C., the area of an isosceles triangle seems to have been found by multiplying onehalf the base by a side instead of by the altitude. This method has been regarded as remarkable on account of its crudity, but when we are told that more than two thousand years later the Roman surveyors were taught to find the area of such a triangle by finding the product of one-half of the numerical measure of the base and a number which is even larger than the numerical measure of another side, there seems to be sufficient ground for surprise even in a scientific matter.

It must be admitted that the instances cited above are insufficient to establish the fact that a general history of science is an unusually favorable ground for the breeding and the propagation of scientific errors. In fact, all that has been attempted here is to advance a few reasons why one might suspect danger here, and to support these reasons by illustrations which were assumed to be also of interest to the reader on account of their unusual intrinsic features. Perhaps a more conclusive argument in support of the thesis in question is furnished by the fact that G. Eneström noted more than two thousand desirable changes relating to the general history of mathematics by M. Cantor, to which reference was made.

THE BIOLOGY OF DEATH. III-THE CHANCES OF DEATH'

By Professor RAYMOND PEARL

THE JOHNS HOPKINS UNIVERSITY

1. THE LIFE TABLE

TP to this point in our discussion of death and longevity we have,

UP

for the most part, dealt with general and qualitative matters, and have not made any particular examination as to the quantitative aspects of the problem of longevity. To this phase attention may now be directed. For one organism, and one organism only, do we know much about the quantitative aspects of longevity. I refer, of course, to man, and the abundant records which exist as to the duration of his life under various conditions and circumstances. In 1532 there began in London the first definitely known compilation of weekly "Bills of Mortality." Seven years later the official registration of baptisms, marriages and deaths was begun in France, and shortly after the opening of the seventeenth century similar registration was begun in Sweden. In 1662 was published the first edition of a remarkable book, a book which marks the beginning of the subject which we now know as "vital statistics." I refer to "Natural and Political Observations Mentioned in the Following Index, and made upon the Bills of Mortality" by Captain John Graunt, Citizen of London. From that day to this, in an ever widening portion of the inhabited globe we have had more or less continuous published records about the duration of life in The amount of such material which has accumulated is enormous. We are only at the beginning, however, of its proper mathematical and biological analysis. If biologists had been furnished with data of anything like the same quantity and quality for any other organism than man one feels sure that a vastly greater amount of attention would have been devoted to it than ever has been given to vital statistics, so-called, and there would have been as a result many fundamental advances in biological knowledge now lacking, because material of this sort so generally seems to the professional biologist to be something about which he is in no way concerned.

man.

Let us examine some of the general facts about the normal duration of life in man. We may put the matter in this way: Suppose we started out at a given instant of time with a hundred thousand infants.

1Papers from the Department of Biometry and Vital Statistics, School of Hygiene and Public Health, Johns Hopkins University, No. 30.

equally distributed as to sex, and all born at the same instant of time. How many of these individuals would die in each succeeding year, and what would be the general picture of the changes in this cohort with the passage of time? The facts on this point for the Registration Area of the United States in 1910 are exhibited in Figure 1, which is based on Glover's United States Life Tables.

FIG. 1. LIFE TABLE DIAGRAM. FOR EXPLANATION SEE TEXT

In this table are seen two curved lines, one marked l, and the other d. The l, line indicates the number of individuals, out of the original 100,000 starting together at birth, who survived at the beginning of each year of the life span, indicated along the bottom of the diagram. The dy line shows the number dying within each year of the life span. In other words, if we subtract the number dying within each year from the number surviving at the beginning of that year we shall get the series of figures plotted as the l, line. We note that in the very first year of life the original hundred thousand lose over one-tenth of their number, there being only 88,538 surviving at the beginning of the second year of life. In the next year 2,446 drop out, and in the year following that 1,062. Then the line of survivors drops off more slowly between the period of youth and early adult life. At 40 years of age, almost exactly 30,000 of the original 100,000 have passed away, and from that point on the l, line descends with ever increasing rapidity, until about age 80, when it once more begins to drop more slowly, and the last few survivors pass out gradually, a few each year until something over the century mark is reached, when the last of the 100,000 who started so blithely across the bridge of life together will have ended his journey.

This diagram is a graphic representation of that important type of document known as a life or mortality table. It puts the facts of mortality and longevity in their best form for comparative purposes. The

first such table actually to be computed in anything like the modern fashion was made by the astronomer, Dr. E. Halley, and was published in 1693. Since that time a great number of such tables have been calculated. Dawson fills a stout octavo volume with a collection of the more important of such tables computed for different countries and different groups of the population. Now they have become such a commonplace that elementary classes in vital statistics are required to compute them.

2. CHANGES IN EXPECTATION IN LIFE

I wish to pass in graphic review some of these life tables in order to bring to your attention in vivid form a very important fact about the duration of human life. In order to bring out the point with which we are here concerned it will be necessary to make use of another function of the mortality table than either the l, or d, lines which you have seen. I wish to discuss expectation of life at each age. The expectation of life at any age is defined in actuarial science as the mean or average number of years of survival of persons alive at the stated age. It is got by dividing the total survivor-years of after life by the number surviving at the stated age.

In each of the series of diagrams which follow there is plotted the approximate value of the expectation of life for some group of people at some period in the more or less remote past, and for comparison the expectation of life either from Glover's table, for the population of the United States Registration Area in 1910-the expectation of life of our people now, in short-or equivalent figures for a modern English population.

Because of the considerable interest of the matter, and the fact that the data are not easily available to biologists, Table 1 is inserted giving the expectations of life from which the diagrams have been plotted.