1.1. Review by: L von Bortkiewicz.
Journal of Economics and Statistics 52 (3) (107) (1916), 421-424.

This paper is primarily devoted to the question of how the population of a country or city can be determined for a point in time between two censuses based on the results of these two censuses. This can, of course, only be done approximately, based on certain assumptions about the population trend within the period in question. This gives rise to various approximation methods, of which the author examines the following three in detail. He calls them "arithmetic," "geometric," and "harmonic" because they can be related to the concepts of the arithmetic, geometric, and harmonic mean, respectively. The first two methods have been in use for a long time, and their mutual relationship is expressed in the fact that, while the former uses linear interpolation between the population numbers themselves, the latter uses linear interpolation between the logarithms of the population numbers. As for the third, the "harmonic" method, the author arrived at this method independently. Admittedly, Corneille L Landré had already proposed a very similar interpolation method based on the harmonic mean in 1903; however, as he states under "Errata," the author only became aware of it subsequently. Using two examples, one relating to the German Empire (1875-1905), the other to Switzerland (1860-1910), Gumbel shows that the harmonic method yields better results (i.e., results that are more consistent with the actual course of the numbers) than the geometric method, not to mention the arithmetic method. He does not give the reason for this. Yet this reason is easily found by taking a closer look at the structure of the relevant formulas. It becomes clear that the harmonic method must offer an advantage, particularly in cases where the growth rate exhibits an upward trend over the period in question. This is true in the examples used by the author. However, if he had carried out the same calculation, for example, for the United States of America (1870-1900), he would have arrived at the directly opposite result: the geometric method would have won.

2.1. Review of 1980 reprint by: Gabriele Jäger and Stefan Trudewind.
Kritische Justiz 18 (1) (1985), 83-87.

On 5 August 1932, Heidelberg statistics professor Emil Julius Gumbel was stripped of his teaching license because he had said at an internal event of the socialist student group on 25 May 1932: "For me, the memorial to war is not a scantily clad virgin holding a palm of victory; the horrors and suffering of war are much better embodied by a swede" (Conspirators, Foreword, p. V).

The figure of Emil Julius Gumbel represents the tradition of democratically-republican intellectuals whose commitment to preserving the Weimar Republic has so far received too little attention. Thus, Emil Julius Gumbel has increasingly been forgotten. When he died in New York on 10 September 1966, no German newspaper was willing to publish an obituary. For this reason, the merit of the small Heidelberg publishing house "Das Wunderhorn," which took the financial risk of reissuing two of Gumbel's books, is to be emphatically acknowledged.

Emil Julius Gumbel was born in Munich in 1891 to Jewish parents. In his birthplace, Gumbel studied mathematics and economics. Shortly after receiving his doctorate, he volunteered for military service at the outbreak of the First World War. His patriotism was quickly dampened by the horrors of war, and Gumbel, a volunteer soldier, became an opponent of the war and joined the USPD (Union of German Social Democratic Party). However, his beliefs were less characterised by his membership in the USPD and later the SPD than by his decades-long commitment to the "German League for Human Rights." This association of democratic-republican pacifists remained the true political home of the bourgeois intellectual Gumbel. In 1923, Gumbel habilitated in Heidelberg, where he taught at the university until the withdrawal of his venia legendi forced him into early exile. As early as 1932, he had received an invitation to give guest lectures at the University of Paris, after which Gumbel taught at the Mathematical Institute of the University of Lyon until 1940. In Germany, his writings were burned at the Nazi funeral pyre, and his name was on the new rulers' first expatriation lists. From 1940 onwards, Gumbel lived in the USA and initially found a position at the New School for Social Research in New York. He then taught and researched at Brooklyn College and, finally, from 1953, at Columbia University in New York.

Which of Gumbel's political actions and writings preceded the witch hunt against him, which culminated in the revocation of his venia legendi in 1932? In 1919, Gumbel published his first political pamphlet, "Four Years of Lies," in Berlin for the "German League for Human Rights." This pamphlet was a reappraisal of Gumbel's wartime experiences and a reckoning with the Wilhelminian system. As early as 1924, Gumbel's pacifist commitment prompted Heidelberg University to file a petition to revoke his venia legendi. The decisive factor was Gumbel's following statement. On 26 June 1924, at a rally with the motto "Never Again War," he said he asked those present to rise in honour of the dead, "who, I will not say, fell on the field of dishonour" (Four Years of Political Murder, Preface, p. V). On 16 May 1925, the Faculty of Philosophy decided not to file the application for withdrawal of teaching authorisation after all, since it had not been possible to challenge Gumbel's academic qualifications.

Heidelberg University, which at the time had a reputation for being decidedly republican and liberal, has so far felt unable to comment on the Gumbel case, let alone rehabilitate him. Thus, a recently published catalogue on the work of philosopher Karl Jaspers at Heidelberg University omitted a critical examination of a report Jaspers had prepared on Gumbel in 1924. Unlike the nationalist experts, Jaspers did not doubt Gumbel's academic credentials, but he did question his moral integrity, which was all the more serious given that the psychiatrist Karl Jaspers was also considered a leading figure.

In June 1921, the Berlin-based Malik publishing house published the pamphlet "Two Years of Murder." Carefully researched, this book contains a collection of documents and reports on the political murders committed in Germany since January 1919. Gumbel's sources included court records, verdicts, decisions regarding the dismissal of cases, witness statements, communications from lawyers, relatives of the bereaved, and newspaper articles. In the foreword to "Four Years of Political Murder," Gumbel writes: "I studied the trial reports primarily in right-wing journals." Gumbel compiled tables comparing the murders committed by the political left with those committed by the right, as well as the ratio of the sentences to each other. His result: 22 murders committed by the left were punished with an average of 15 years in prison, whereas 332 murders committed by the right were punished with an average of only four months. These figures illustrate the kind of republic that was being protected here. "Four Years of Political Murder" is the expanded fifth edition of "Two Years of Murder" and was published in 1922. It contains numerous new cases. Separate chapters list the cases from the murder of Eisner to the fall of the Bavarian Soviet Republic, as well as the murders related to the Kapp Putsch. The new edition of "Four Years of Political Murder" was expanded with a memorandum from the Reich Minister of Justice and a foreword by Hans Till, who briefly describes Gumbel's biography.

In July 1921, then Reichstag member Gustav Radbruch had called on Reich Minister of Justice Heinze to investigate the cases presented by Gumbel. When Radbruch himself became Reich Minister of Justice, he pushed the investigations with the relevant state governments of Prussia, Bavaria, and Mecklenburg. After Radbruch's resignation in October 1923, the Reichstag's memorandum on Gumbel's murder statistics was published a month later - remarkably, in only a single copy! At his own expense, Gumbel was given a copy, which he published in 1924 with the Berlin Malik publishing house. In the collections of material "Two Years of Murder" and "Four Years of Political Murder," Gumbel had attempted to let the facts speak for themselves. In his commentaries on the memorandum, however, he no longer holds back his own opinion about the murders and their causes. He particularly denounces the laxity with which the Bavarian memorandum deals with legal issues. The Prussian and Mecklenburg memorandums at least attempt to provide legal justification for the verdicts acquitting right-wing murderers. "Bavaria does not need this. Here, right-wing murders are simply registered, everyone receives a few lines, and it always simply states: 'The proceedings have been discontinued,' or even more briefly, 'The perpetrators could not be identified'".

The Mecklenburg memorandum deals exclusively with the murders committed by the Kapp Putschists. In this context, Gumbel refers to the first appearance of secret societies and their bloody methods. While in earlier cases, the murderers were evaded punishment by delaying investigations or not opening the main proceedings, here an even more effective weapon was available: the amnesty. This was originally enacted to protect thoughtless followers from punishment; however, through a completely one-sided interpretation of the concept of "Führer," it was possible to exempt even leading generals from responsibility, since they were all acting in execution of the orders of Kapp, Lüttwitz, and Lettow Vorbeck.

The leniency with which the Kapp-Putschists were prosecuted stood in stark contrast to the rigorous punishment of the Munich Soviet Republicans. Murders of left-wing politicians mostly went unpunished; leads that could have helped solve the crimes were often ignored by the investigating authorities. "The sense of justice was subordinated to the fight against communism." Gumbel does not necessarily accuse the Weimar judges of consciously perverting the law. He writes: "They lack an awareness of the criminality of their actions. From the old days, when today's economic system was absolutely protected from external attacks and when the supporters of the right-wing parties undisputedly formed the upper classes, the idea that a number of murderers and murderers could emerge from this caste is inconceivable to them." Gumbel had believed that his accusation against the German judiciary for failing to prosecute over 300 political murders could only have two effects: "Either the judiciary believes I am telling the truth, in which case the murderers will be punished. Or they believe I am lying, in which case I will be punished as a slanderer. In fact, a third, completely unforeseen thing has happened... not a single attempt has been made by the authorities to dispute the accuracy of my allegations. On the contrary, the highest competent authority, the Reich Minister of Justice, has explicitly confirmed my allegations several times. Nevertheless, not a single murderer has been punished".

While, according to Gumbel's statistics, the Republican Council members were sentenced to a total of 5,000 years of imprisonment, even leading Kappists remained unpunished and continued to threaten the existence of the Republic through their activities. "Four Years of Political Murder" documents the highly active role of the judiciary in the dissolution of the Weimar Republic.

In addition to statistics on political murders, Gumbel's book also includes a chapter entitled "On the Sociology of Political Murders." In it, he describes, among other things, the social and organisational structure of nationalist secret societies in Germany for the first time; a description he then continues in his book "Conspirators."

3.1. Review of 1979 reprint by: Gabriele Jäger and Stefan Trudewind.
Kritische Justiz 18 (1) (1985), 83-87.

The book "Conspirators: Contributions to the History and Sociology of German Nationalist Secret Societies since 1918" was first published in March 1924 by Malik Verlag. The foreword was written by A Freymuth, then President of the Senate at the Berlin Higher Regional Court. A Freymuth was one of the few republican-minded judges, and it is noteworthy that, as a representative of the judiciary, he endorsed Gumbel's theses.

The new edition of this work features an informative foreword by Kann Buselmeier, in which she evaluates the results of her extensive research on Gumbel and his publications. Furthermore, the publisher "Das Wunderhorn" has published the "Decision of the Heidelberg Faculty of Philosophy of May 16, 1925, in the matter of Privatdozent Dr Gumbel" as well as a protest statement by republican and socialist university professors in response to renewed, sharp attacks against Gumbel in 1931. Both documents provide insight into the way in which inconvenient critics of the regime were treated at that time.

While Gumbel has demonstrated the one-sidedness of political justice in the Weimar Republic in his previous writings, his main concern here is to examine the organisational structures of the nationalist secret societies. In one of the chapters, Gumbel compiles a list of the more well-known nationalist organisations. He describes their main goals and internal structure and names the leaders. He traces their sources of funding and assesses their membership structure; overall, Gumbel provides a lexical overview of the mischief of the nationalist secret societies. Further sections deal with, among other things, with the Kapp Putsch and Hitler's attempted putsch of November 9, 1923, the collaboration between the Reichswehr and secret societies, and, last but not least, the special situation in Bavaria. The compilation of this wealth of detailed information was based on the exemplary nature and sensational nature of the incidents. Names, abbreviations, and events that were referred to daily at the time were to be named and explained in their context. Gumbel wanted to warn against organisations that appeared harmless and provide assistance with political orientation.

The facts and documentary reports are only sparsely commented on, and a specific evaluation analysing individual cases, for example, for their pioneering role, is lacking. However, Gumbel provides insights into details that are often neglected in summary analyses.

The collected facts sound so scandalous that one wonders today why Gumbel himself was not the victim of an attack. Gumbel did indeed receive several death threats, but hardly any from members of the organisations he directly attacked in his books. Klaus Theweleit therefore suspects that those named were filled with narcissistic pride that someone was researching their activities so carefully. However, one should not overlook the fact that Gumbel's revelations endangered the nationalist secret societies less than he had hoped, for his writings also demonstrate that nationalist activities themselves were hardly combated, but rather encouraged, at the highest levels. Gumbel described this attitude with blunt realism: "In all these cases, the judiciary, under the influence of representatives of the ancien régime, instinctively saw the murdered as their enemy, but in the murderers as members of good society, which it was their task and duty to protect". The reports also demonstrate the official absolution of nationalist activities, including murder and coups.

Gumbel explores the causes of this fatal and pervasive anti-republican sentiment in his final chapter, "Principles of the Secret Societies." Two of his theses are particularly noteworthy: There was no self-evident national feeling in Germany, but at most a strong bond with the individual monarchical dynasties. Although these disappeared irrevocably with the end of the First World War, the population's personal fixation on their royal houses persisted. Drawing on socio-psychological arguments, Gumbel concludes: "Deprived of its object, this love transforms into hatred of the current system". The second thesis states that in the 1920s, the social fabric in Germany was disrupted. The most decisive result of this restructuring process was the relative impoverishment of the old middle class. It was precisely the middle class that was most affected by the policies of the Reich. It was the middle class that suffered the most from inflation, and furthermore, the former professional ethos, the honour of civil servants, lost importance in the Republic. Gumbel maintains that as a result, those who were supposed to be the intellectual pillars of the Republic became its enemies.

With diligence and meticulousness, Gumbel exposed the Weimar judiciary's unwaveringly anti-republican attitude in its jurisprudence. The results are clear: The practice of the German courts "is anything possible, but justice is not." What is surprising is how early the tendencies toward fascism became apparent; that even at the beginning of the 1920s, it is evident how little the SPD doubted the authorities of the time, and how criminally neglected a personnel policy committed to the republic was in the Reichswehr and throughout the state and administrative apparatus. This sheds light on an understanding of politics that focused solely on the majority in the Reichstag. The fixation on the highest state offices went hand in hand with a fatal undervaluation of the bureaucracy and the lower political levels.

While there were many who warned that the Weimar Republic was not dedicated to the advancement of democracy, but rather to the resurrection of monarchist and nationalist forces, hardly anyone researched this as meticulously as Gumbel. His studies on the judiciary appear to Fleute as indicators of a social development that logically culminated in National Socialism. Kurt Tucholsky judged the book "Two Years of Murder" to be the "most important publication of the last three years." He wrote: "The trust of decent people in the political German jurisprudence is now probably finally over."6 This hope was dashed. The Reichstag felt compelled to confirm Gumbel's findings without initiating a single countermeasure. Gumbel did not know how to deal with this arrogance of the rulers. He was the type of intellectual who distanced himself from everyday political life. He believed that something would change in politics if only enough evidence was gathered carefully. Gumbel's work was the tragedy of an Enlightenment thinker who believed in the power of reason, who, while able to analyse the trends of his time very precisely, failed to grasp the mechanisms of tyranny.

4.1. Review by: R A Freymuth.
Die Friedens-Warte 30 (2) (1930), 62.

Gumbel, a university professor in Heidelberg, is arguably the most resolute writer in waging the struggle against the forces that, even after the fall of the German Empire, the loss of the World War, and the political upheaval of 9 November 1918, seek to maintain the old, time-tested, "national," loyal German spirit in the new republican Germany. Given the current situation in Germany, he is unfortunately compelled to wage war against militarism, against the black and, to a large extent, also against the grey Reichswehr, against political jurisprudence in Germany, against the administration of justice in political matters, against many other things that should actually be different in the other Germany. One should recall Gumbel's important books, "Four Years of Political Murder," "Conspirators," and the memorandum of the Reich Minister of Justice, which was published not by him but by Gumbel. His new book, "Traitors Suffering from Feme," has just been published. He edited this book together with Berthold Jacob and Ernst Falck. The book provides a truly shocking, comprehensive account of the Feme murders, precisely because of its calm and objectivity, and reveals, as far as clarity is possible, the connections that exist and existed between the Feme murders and the black and grey Reichswehr.

Read this book! Its content and value are best clarified by Gumbel's concluding words, which are reproduced here in abbreviated but true-to-word and meaningful form:

"The Black Reichswehr has been disbanded; the Feme has disappeared. The political murders have ceased. Some murderers have even been punished. Thus, all our considerations are outdated. ... No: The most significant murderers are walking free. Black Reichswehr and Feme murders continue to work - with other means. The Feme was followed by the bomb. ... Where the bullets of the Feme murderers once rang out, the treason clauses now resound as an effective substitute... The draft penal code, enacted into law, will realise the old goals with entirely legal means. Germany will stand in the shadow of the treason clauses. ... And the working class should know: If the current triumph of capitalism falters, and if democracy is no longer sufficient to protect the ruling class, the Black Reichswehr and Feme murders will rise again. If this time remembers our work, it has not been in vain."

5.1. Review by: Pierre Depoid.
Annales sociologiques. Série E. Morphologie sociale, langage, technologie, esthétique (3/4) (1942), 58-59.

This is a mathematical study devoted to mortality at advanced ages and intended to determine the largest value of an unlimited distribution corresponding to a given number of observations. Various mathematicians have put forward the hypothesis that the duration of human life is one of the constants of biometric functions and that its determination is linked to a special analytical hypothesis on the increase in mortality with age: at this limit age, the survival function would be zero and undefined at higher ages. The author of the present work starts from the hypothesis that the survival function is everywhere continuous and approaches zero only asymptotically; using the calculus of probabilities, he deduces the last age corresponding to a given number of observations. Starting from the normal age (dominant of the distribution of deaths) located around 75 years, it studies the distribution of deaths according to age assumed to be unlimited in the direction of high ages: the calculation is developed using two remarkable survival functions, that of Lexis which consists of assuming that the normal age is the average of a Gaussian distribution, and that of Gompertz which assumes that the intensity of mortality increases as an exponential function of age. The author distinguishes between the oldest age, a statistical variable relative to a given number of observations and whose distribution depends on this number, and the last age, the most probable value of this distribution. On the other hand, the oldest age tends towards a final value when the number of observations increases: the limit age will be considered as the mathematical expectation of the distribution envisaged.

Tables have been drawn up giving, for the Lexis and Gompertz formulas, the values ​​of the last age, the upper age, and the standard deviation used to define the highest age according to the number of observations. These theoretical results are compared with the numerical values ​​provided by the American mortality tables for 1901–1910 and by deaths in Sweden. Furthermore, for Switzerland and Sweden, the author uses observations from the last fifty years to study the distribution of the highest observed values. In all cases, the agreement between theory and observations is satisfactory despite the asymptotic form of the functions considered; they approach zero so closely that we practically arrive at a upper age, because, to increase this age by one year, the number of observations must be multiplied by a factor that increases very rapidly as the age increases.

5.2. Review by: K V I.
Current Science 7 (5) (1938), 244.

This book deals with the problem of determining the extreme duration of human life by the study of a mortality table. It is well known that the mortality tables cannot give valid results in advanced ages. Until now the books dealing with this subject introduce a discontinuity in order to avoid this difficulty. But as Steffenson has remarked this raises logical difficulties. The author avoids this difficulty by treating the corresponding biometric function as continuous at the same time assuming that it will approach zero in some asymptotic way.

5.3. Review by: W S.
The Mathematical Gazette 22 (251) (1938), 422.

During recent years there has been an increasing interest about the duration of human life and the possibility of a continuation of the improvement that has taken place owing, it is assumed, to an increase in our sanitary and medical knowledge. Many formulae have been constructed to express the life curve, the best known and most used being those by Lexis and what is known as the Gompertz-Makeham curve, which are the formulae chiefly used by M Gumbel in estimating the final duration of human life. According to these formulae, the age of Methuselah (969 years), although extremely improbable, would not be impossible.

M Gumbel has also made estimates from data given by some American tables, giving records for Blacks and Whites, native and foreign, and also from Swedish and Swiss statistics. For Sweden he gives as the final age for the year 1932 as 105.79, and estimates the final age for the year 2036 as 106.49. Amongst American records we find the oldest male was a negro, 135; the oldest white, 119. The chief difficulty in this matter is the absolute unreliability of the records. We have in this country rare cases, for we find according to documents printed in the Royal Society Transactions the case of Thomas Parr, who is said to have lived 152 years 9 months, and Henry Jenkins, 169 years. It is difficult to see how reliable records could be obtained for any period over 100 years, and all such records must be suspect. The paper by M Gumbel is well worth the attention of all those interested in statistical questions.

6.1. Review by: Albert Salomon.
American Sociological Review 4 (2) (1939), 304-305.

A number of excellent books on the conditions of life in Germany, in particular on that of the German universities under the Nazi regime, have been written in the U. S. A. by American scholars. Nevertheless, this volume, the collective work of a group of exiled German scholars, takes a place of its own. For it gives evidence to the fact that actual experience contributes to a complex interpretation of a bewildering and embracing situation. The work supplies new information on how the scientific disciplines were impregnated with the Nazi philosophy, so as to make even physics and especially biology "German." The volume carefully analyses from various and different viewpoints the political and pseudo-religious ideologies of the ruling party. In particular Lieb's excellent analysis of Rosenberg's philosophy as the expression of the intellectual and moral nihilism of the present regime throws much light on the true character of the political system. Finally, the work will stimulate reflection on the sociological structure of this emigration, which is completely different from the flight of the French aristocrats in the time of the revolution and from the European emigration consequent to the breakdown of the 1848 rising. This reflection is likely to lead to the discernment that we have before us not really an emigration, but a case of mass expulsion. It will then be seen that unlike the case of earlier emigration, these "emigrants" are, with negligible exceptions, not held together by an unifying impulse, but only by the fact that they are opposed to and persecuted by the powers ruling Germany to-day. This totalitarian expulsion is not the result of a political revolution like those of the past, but is rather the consequence of an absolute religious fanaticism. Hence it is hard to find an integrating idea among the various groups and stages of this new migration. It is not without significance in this connection that the most thought-provoking chapter in this book is the contribution of a theologian. The book presents a clear picture of the various stages of this new migration and of the heterogeneity in attitude, mode of thinking, and intensity of these migrants. These differences, however, are so well bridged by the common immediate experience of a new situation that it prepares the foundation for a new solidarity which may help to bring about a new pattern of humanity.

6.2. Review by: E Y Hartshorne.
American Journal of Sociology 44 (4) (1939), 578-579.

The first concerted effort on the part of German scholars now in other countries to discuss in their own language the topic "National Socialism and Science," this volume opens with a list, "certainly incomplete," of thirteen German university teachers who have committed suicide since 30 January 1933, and one who has been murdered. It closes with brief autobiographical notices by each of the fifteen contributors to the present work. These are the ones who will "carry on," so to speak, since the implication of the whole volume is that the traditions of German science survive only outside Germany. The notices indicate that of the thirteen contributors who have left their country "since Hitler" at least four are "Aryan," two of these being World War veterans. Six state as their field of work the social sciences, two more law, one education, one Scandinavian literature, one theology, one biology, and one mathematics and physics.

Arresting both for what it is and for what it says, the book as a whole may be viewed as an effort on the part of this group of exiled scientists to re-create a sense of the erstwhile community of scholarship which bound them together. In this respect the volume has an apologetic aim - namely, to indicate that, whatever may have been happening in the home-land, German science still exists.

Interpreting this aim, each in his own fashion, some of the contributors eschew political reference altogether, seeking only to give expression in a concrete piece of work to what they hold to be the genuine spirit of German scholarship, as, for example, in Julius Lips's study of mutual aid among the Labrador Indians and in Walter Landauer's presentation of certain recent developments in the field of genetics. Others, like the physicist Gumbel and the biologist Schaxel, pursue a policy of direct attack on what goes under the name of "National Socialist Science." The rest, with varying, degrees of success, attempt to "explain" various aspects of the contemporary German "situation." While it is sometimes difficult in reading these sections to decide whether a given statement derives more from a disinterested desire to interpret the facts or from an irrepressible drive to justify the writer-which in one instance leads to the conclusion that the scientists in question "have been ejected not merely by Hitler, but by the whole of that Germany which, since I864, has climbed to power on the shoulders of a victorious Prussia," nevertheless in some cases, with the elements of diatribe and self-vindication discounted, the discussion is singularly illuminating.

Especially noteworthy are Theodor Geiger's essay on "The Role and Destiny of Intellectuals"; J Schaxel's "Fascistic Falsification of Biology," which suggests that one of the chief "uses" of the racial ideology is the promotion among those who accept it of an attitude of political fatalism; and finally E J Gumbel's able and witty exposition of some of the intellectual paradoxes and personal conflicts involved in the attempt to establish an "Aryan Natural Science."

6.3. Review by: L L Snyder.
The Annals of the American Academy of Political and Social Science 202 (1939), 255.

The catastrophic decline of German culture and science in favour of a higher emotionalism, and the substitution of guns for higher learning in the present German Reich, have not as yet resulted in the eclipse of German mentality. Much notable work is still being produced by Germans-outside of Germany, abseits von der Reichskulturkammer. In the December 24 (1938) issue of Das Neue Tagebuch, published in Paris by emigrants, there are listed nearly two hundred books written by exiled Germans in 1938, by the Manns, the Zweigs, and the best names in German literature and science.

The present book is a striking indication of fact that the Weimar mind persists in spite of the spirit of Munich. Dedicated to "Free Science," the collection includes the work of some of the most important scholars in pre-Hitlerian Germany. Anna Siemsen writes on problems of education; Theodor Geiger on tasks of the intellectual; F W Förster on the tragedy of German Christians; Fritz Lieb on National Socialist nihilism; Julius Lips on public opinion among the Indians of Labrador; Gottfried Salomon on law in Germany; Arthur Rosenberg on the task of the emigrant historian; Julius Schaxel on fascist falsification of biology; and E J Gumbel on "Aryan" natural science.

Although the contents are arranged into three major sections - spiritual situation, state and society, and natural science - the collection is somewhat haphazard in content and arrangement. This by no means detracts from the value of a collection of this kind. The purpose of the book is not to satirise and condemn the present tendencies in Germany, but rather to help fashion the foundations for polemics and criticisms which are valueless without scientific support. The authors represented in the present volume have demonstrated that the old German culture will persist in spite of temporary efforts to stifle it.

A unique feature of the collection is extended biographies and bibliographies of the works of all authors represented. This excellent plan should be adopted by editors of all collections of essays. At the beginning of the book is placed a list bordered in black of noted German professors who were murdered, died by suicide, or died "apparently by suicide." A book of this kind, testimony to the fact that their spirit lives on, is perhaps the best type of monument to those German scholars whose lives have been lost in post-Weimar Germany.

7.1. Review by: B C Brookes.
The Mathematical Gazette 39 (330) (1955), 341-342.

In its development classical statistical theory has been mainly concerned with the properties of averages and in its applications with the average properties of populations and materials. But in many practical problems, especially in engineering design, it is the extreme rather than the average value of a variate which may be the determining factor, and, so far, statistical theory has not provided the engineer with a general theory of extreme values. This omission may well be one of the reasons for the reluctance of engineers, in general, to adopt statistical techniques in solving design problems.

The strength of a chain depends not on its average strength, but on the strength of its weakest link; an aircraft has to be designed to withstand the stresses not of the average conditions but of the extreme conditions it is likely to meet. The materials which mechanical and electrical engineers have to use are not homogeneous; often, as with insulating materials, it is the expected minimum value of some physical quantity on which the engineer has to base his design. In the past, problems of design have been solved by various ad hoc methods; the engineer has usually based his calculations on average values of the strengths of his materials and has then confessed his ignorance and doubt by multiplying his answer by the arbitrary number he euphemistically calls the "safety factor". The author of the pamphlet under review has spent many years establishing a statistical theory of extreme values and he has succeeded in providing such a theory in a form which is readily applicable to practical problems.
...
The pamphlet consists of four lectures in which the author surveys the general problem, describes some graphical methods (using a special form of probability graph paper), outlines the mathematical theory, and finally describes some applications. His attitude is that of the engineer rather than of the mathematician; his exposition of the mathematical theory, limited in any case to one lecture, is therefore cursory. Though we must be grateful to the American National Bureau of Standards for sponsoring this publication we hope that the author will not be deterred from presenting his work in a more considered and expanded form. The statistical theory of extreme values could give a more logical basis both to many aspects of engineering testing and design and also to theories of the breakdown of materials under stress. It opens a new field of exploration for mathematical statistics, in which the author indicates some unsolved problems, and provides computers with some new functions to tabulate (though some tables have already been published in No. 22 of this series). It may also contain some disturbing implications for the theory of statistical inference. This slim pamphlet is likely to be of far-reaching importance.

7.2. Review by: K D Tocher.
Journal of the Royal Statistical Society. Series A (General) 118 (1) (1955), 106.

Many readers of the statistical journals will have noted the steady stream of papers by E J Gumbel concerning the distribution of extreme values, and many will, like myself, have mentally docketed these as of theoretical value to be read when the need arises.

The National Bureau of Standards has now reproduced a set of lectures delivered by Gumbel which summarises the results of a life-time's work. The first lecture gives the motivation behind his researches and shows that the practical problems of flood prediction and allied phenomena lead directly to a study of the relation between the distribution of individual values, the number of values involved and the distribution of extreme values. It also gives an interesting historical survey.

The second lecture develops the law of rare occurrences, introduces the concept of the return period and discusses the determination of these from practical data by the use of specially ruled paper.

The third lecture derives the exact distribution of extreme values and asymptotic expressions for them. A theory of types of initial distribution is developed wherein the limiting extreme distribution does not depend on the exact nature of the distribution of initial values, but only on its type, in particular, on its behaviour for large values of the independent variate.

The fourth lecture is devoted to applications.

8.1. From the Preface.

The theory of extreme values is mentioned in most of the recent textbooks of statistics. However, this seems to be the first book devoted exclusively to these problems. It is meant for statisticians, and statistically minded scientists and engineers. To spread the application of the methods, the author has tried to keep it on an elementary level. Graphical procedures are preferred to tedious calculations. Special cases and easy generalisations are given as exercises. Therefore, the book may even meet the requirements of a textbook.

Since the line of thought pursued here is new, the reader should not expect complete solutions. There are bound to be shortcomings and gaps in a first treatment. Despite the usual prescriptions of academic habits, unsolved problems are here clearly stated. Perhaps a carping critic might maintain that more problems are raised than solved. However, this is the common fate of science.

This book is written in the hope, contrary to expectation, that humanity may profit by even a small contribution to the progress of science. The opportunity to start this book presented itself when the necessary free time was imposed upon the author. A first draft of the manuscript was written in 1949 under a grant by the Lucius N Littauer Foundation. This draft was revised in 1950 under a grant from the John Simon Guggenheim Memorial Foundation. The work continued while the author was engaged as a Consultant at Stanford University and later as an Adjunct Professor at Columbia University, partially under a grant from the Higgins Foundation, and under contract with the Office of Ordnance Research. Finally, a further grant from the Guggenheim Foundation made it possible to engage Professor Frank Lee (Columbia University) for the drawing of the graphs. The author wishes to state his sincere appreciation to these agencies for their help, without which he would not have had the chance to finish this book.

8.2. Review by: F N David.
Biometrika 47 (1/2) (1960), 209.

A saying, to be found equivalently in many languages besides English, states that 'a chain is as strong as its weakest link'. This aphorism states succinctly and accurately why statisticians, and especially perhaps statisticians advising engineers, are concerned with extreme value problems. E J Gumbel has been concerned for many years with the study of these and allied problems and has, in this present book, put down the fruits of his researches. The book, written by a specialist, is therefore one which would a priori command attention, and the contents discover to us that this attention is very much worth while.

Prof Gumbel requires of his readers that they should have as prerequisites some calculus, some analysis and a little more than a smattering of mathematical statistics and probability. As delineated by the author the mathematical structure of the book is not difficult but the uninitiated will not be able to plunge into it in a carefree way. The book begins with a longish chapter (38 pages) in which the author sets out the tools he is going to use and seeks to orientate his readers so that they understand the blue-print from which he is going to work. Chapter 2 is on order statistics and Chapter 3 on the exact distribution of extremes. As is well known the study of exact distributions in any form of 'order' problem is, on the whole, an unprofitable proceeding so we turn in Chapter 4 to the analytic study of extremes, with asymptotic results following in Chapters 5, 6 and 7. The range has Chapter 8 to itself. There are 44 tables and over 100 graphs.

The emphasis, as far as the author's examples go, is on mechanical engineering problems (such as fatigue in metals) and on climatic difficulties which are really civil engineering problems (such as floods and droughts). But it would be a mistake to conclude that only engineers with a statistical slant could profitably read and learn. Anyone who is interested in extreme values will have to read this treatise in order to find out what has been done. In the limited field of mathematical statistics which this book delineates it is encyclopaedic and can be unhesitatingly and unreservedly recommended.

8.3. Review by: G S James.
Journal of the Royal Statistical Society. Series A (General) 122 (2) (1959), 243-245.

"A book should have either intelligibility or correctness. To combine the two is impossible." Thus the quotation at the head of the final summary in Professor Gumbel's book. The reviewer feels that this book errs on the side of correctness, but he writes as one not very familiar with the subject. The chief impediments to a ready understanding are an over-condensed literary style, and the mathematical notation. The author has obviously attempted to systematise the latter: that he has not been entirely successful is probably not his fault. One feels, however, that this is a case where a list of the (varying) meanings of the main symbols used might have proved helpful. On the stylistic side, the desire for conciseness leads to the use of many technical terms and abbreviations, some of which tend to convey a meaning other than that intended. Examples are the "probability" of x rather than its (cumulative) distribution function, and "initial median" for the median of the initial (i.e. parent) distribution. Fortunately the book possesses an adequate index. Another irritating feature arises because all that can be said about largest values can be said, with appropriate changes, about smallest values. But would it not be better left unsaid? Here is an example: "In consequence of (2) and (7), the logarithms of the average maxima (minima) of the smallest (largest) values decrease (increase) and the average maxima (minima) of the smallest (largest) values increase (decrease) as the logarithms of the numbers of extremes increase."

The book is intended for statisticians and statistically-minded scientists and engineers, and the author states that he has tried to write on an elementary level and to use graphical procedures in place of complicated calculations. The first objective has been achieved, in that an undue standard of rigour and generality has not been imposed; the book is definitely on statistics rather than on abstract probability theory. As for the graphs, there are about one hundred of these, nearly all illustrating theoretical relationships rather than practical data. Most of them use transformed scales on one or both axes, and in this case two or three different scales may appear against each axis. A variant is the drawing of more than one curve, to different scales, on the same diagram. Many will feel that the effort of unravelling the meaning of some of these is not worth while. Transformed scales are very useful for plotting experimental data in straight line form, but they hardly help one to get the "feel" of a theoretical relationship when it has been distorted into a straight line. It is unfortunate that the symbols on the diagrams are occasionally different from those in the text.

One of the most useful features of the book is a closely-printed 22-page bibliography of works in the pure and applied field; and selected references to further reading are given at the ends of sections. There are also "exercises" and "problems", the latter being more difficult questions, some of which could serve as research projects. Introductions and summaries are placed at strategic points throughout the book, and should prove helpful in using the book for reference purposes.
...

Statisticians will be grateful to Professor Gumbel for writing a book which will be referred to for many years to come.

8.4. Review by: B F Kimball.
Biometrics 17 (1) (1961), 160-161.

This work is essentially a source book on the statistics of extreme values. Furthermore it is the only source which is comprehensive. Viewed as a source book it is well arranged and well documented. The bibliography includes over 600 books and articles, to which specific references are made throughout the book. Of these, 34 are papers by the author, or written in collaboration.

The first chapter deals with definitions and the basic concepts. In the second chapter, on order statistics, the theory of exceedances is set forth and handled in a straight-forward and concise manner.

In Chapter 3 the exact distribution of the extreme values of a sample are introduced. Chapter 4 is given over to an analysis of the exact distributions of extremes, pointing out the essentially different types which are important.

These first four chapters comprise nearly half the subject matter and include all the material dealing with "exact" distributions of extreme sample values. Perhaps the principal advances in the study of the distribution of extremes in the last twenty five years have been brought about by recognition that the asymptotic forms of these distributions are usually adequate as approximations to the exact distributions in the applied field, and are much simpler to deal with.

The asymptotic forms fall naturally into three classes, of which the doubly exponential is of the most consequence since it is the asymptotic form which results from a universe having an "exponential" type of cumulative distribution function, such as the logistic, normal and Gamma distributions. A second class, or "second asymptote" as referred to by Gumbel, stems from distributions of the Cauchy or Pareto type. The third class is often referred to as Weibull's distribution when used for the lower extreme. Both can be obtained from the first by suitable logarithmic transformations. The first "asymptote" and its applications are discussed in Chapters 5 and 6, covering about 100 pages. This class is of particular interest to biologists - for example, in studying extinction times of bacteria'. The second and third asymptotes are treated in Chapter 7. Chapter 8 is devoted to the asymptotic forms of the range and mid-range, and their applications.

This reviewer finds the author very ingenious in deriving methods which he believes will facilitate application of the theory to practical problems. Some of these ingenious devices are open to criticism. For example, the discussion of the topic Fitting Straight Lines on Probability Papers is more ingenious than logical.

8.5. Review by: G Leti.
Genus 22 (1/4) (1966), 392-393.

This treatise is the fruit of the work, more than thirty years, carried out by the author in the field of the theory of ordinal values ​​and of extremes in particular. He was a precursor in this study, because it is only a short time that extreme values ​​have been re-evaluated in statistics by the theory of samples - given the timeliness of the information that they can provide while previously the interest of statisticians was mainly directed to average values ​​and the most frequent events. With great clarity and completeness the author exposes the contributions brought to the theory of extremes by authors from many countries, by statisticians, by probabilists and also by scholars of engineering and other different subjects (in the bibliography are cited about five hundred publications in English, French, German, Spanish, Russian, in other languages ​​and, something very rare, also in Italian [Note by EFR. This review is in Italian in an Italian journal]).

The author's main aim has been the complete intelligibility of the text, which, at times, as he himself makes clear, he has preferred to the rigour of the treatment: in fact he has tried to present the argument in the most elementary way possible, often resorting to intuitive reasoning and sometimes preferring to use graphs instead of complicated formulas.

The book is divided into eight chapters. The first is dedicated to the exposition of the methods on which the treatment is mainly based, in the second chapter the ordinal values ​​are treated of which the extremes are particular cases. In the third chapter the exact distribution of the extremes is treated in dependence on the generating population and the size of the sample and various characteristics of this distribution are studied. The fourth chapter mainly concerns the distributions of samples obtained from the unlimited generating populations of exponential type and Cauchy type. The three following chapters are dedicated to the three asymptotic distributions of the extremes: the first concerns the distributions relative to the exponential type populations, the second to the Cauchy type populations and the third to the limited populations. In the last chapter some functions of the extremes are treated: their difference, their sum, the quotient and the geometric mean.

The treatise contains examples taken from the most diverse fields such as hydraulic and naval engineering, meteorology, geology, metallography and actuarial mathematics. The author, as he explicitly declares, does not claim to have said the final word on the subject: he signals the existence of defects and gaps in the treatment, since it is the first theorisation of the subject, and of problems not completely resolved. The existence of still open problems shows the vitality of the subject that, even after the publication of the volume in question, has continued to attract the interest of the author who has, since then, published several works on the subject.

8.6. Review by: G J Lieberman.
Journal of the American Statistical Association 55 (290) (1960), 383-384.

Statistics of Extremes is the definitive work to date on this important subject. It is a complete treatise on the subject, bringing together many fundamental studies by scientists working in different fields and in many countries.

The systematic treatment of extreme values starts with methods which are distribution-free. After some initial distributions have been considered, more general methods are developed, which hold for certain types of distributions; finally the asymptotic theory is constructed.

The book comprises eight chapters. Each chapter consists of several sections divided into paragraphs. Each section starts with a statement of the problems. The first chapter introduces those statistical tools which will be frequently used, especially the concept of intensity function, taken from actuarial statistics, and return period, taken from engineering practices.

The second chapter introduces distribution-free methods and culminates in the proof that a forecast of the number of exceedances of the extreme is more reliable than a similar forecast for the median. The third chapter shows the general properties of extremes which do not require the knowledge of the initial distribution; in particular it sets bounds on the increase of mean extremes with sample size.

The fourth chapter is devoted to exact distributions of the extremes. Here the initial distribution, the parameters contained therein, and the number of observations must be known. Two types of initial unlimited distribution are worked out: the usual exponential type, and the Cauchy type, which possesses a longer tail. Since the distributions of the extremes depend upon the behaviour of the initial distribution at large absolute values of the variate, the two types possess different properties with respect to the extremes. A third type consists of certain limited distributions.

As long as small samples are considered, no new parameters enter into the distribution of extremes. This situation changes as soon as large samples are approximated by the three asymptotic theories which are studied in chapters 5 to 7. Here two new parameters appear, which are connected with the intensity function and return period.

The first asymptotic distribution of extreme values valid for the exponential type of initial distributions seems to be the most important. chapter 5 is devoted to the study of this theory. Applications especially to floods are given in chapter 6. The second and third asymptotic distributions valid for the Cauchy type, and for certain limited distributions, are shown in chapter 7. Certain empirical formulae derived by engineers for the breaking strength of materials are special cases or generalisations of the third asymptotic distribution of the extremes.

The last chapter deals with functions of extremes, their sum, difference, quotient, and geometric mean.

The book contains many detailed examples of the use of extreme values in such diverse fields as flood control, dam, bridge, reservoir, and hydroelectric plant construction, mortality tables, metal fatigue and building codes.

The book will be of "extreme value" to statistically minded people who are interested in the theory of extremes. However, it is possible that the book will not reach many scientists who are interested in the subject matter, but are not well versed in statistics. This is unfortunate, since the book is a treatise on the subject and contains material of vital interest to such people. Dr Gumbel is to be commended on bringing together all the important work in such an important field.

8.7. Review by: Ernest Nagel.
Scientific American 200 (4) (1959), 196-197.

This book is a study of the statistical theory of extreme values by the foremost investigator of the subject. The theory deals with two types of question: Does an individual observation in a sample taken from a distribution fall outside what may reasonably be expected? Does a series of extreme values exhibit a regular behaviour? ("Reasonable" and "regular" must of course be defined by an operational procedure.) The oldest problems connected with extreme values arise from floods. Until recently these were treated by purely empirical methods; now, however, the statistical nature of these phenomena is recognised and new methods of analysis derived from the theory of extreme values are coming into use. Among other phenomena to which the theory can be usefully applied are annual droughts, largest precipitation, snowfalls, maxima and minima of atmospheric pressure and temperature, metal fatigue.

8.8. Review by: W R B.
The Incorporated Statistician 10 (2) (1960), 96-97.

This book is a fine example of a great need in the organisation of statistical knowledge - the monograph which gives a comprehensive review of apart of the statistical "tool-box". Such monographs can be of the substantial kind - as is the book under review - where a summary of the known statistical techniques is presented, or it may be of the shorter kind which serves its purpose by providing an informed guide to the relevant literature. Both of these forms will contain the ordinary (indicative) bibliography although this is no longer sufficient by itself. It should be noted that there is a bibliography of approximately 500 entries in the book under review as well as suggestion in every chapter for its use.

The vast coverage of this monograph indicates that it is the coordinated work of an expert of many years standing: Professor Gumbel has, in fact, been working in this field for at least a quarter of a century. Indeed, his vigour is still great since, besides the evidence of recent papers, there are many "exercises" and "problems" given in this book which provide much thought for development. That such development will come is certain because the practical problems which give rise to the study and statistical analysis of extreme values are among the more important in many branches of engineering - both material as well as human.

As is not unusual in a reference book, the style of writing is compressed: but there is liberal use of the graphic approach. Indeed, there is information in some of the diagrams which is not immediately available elsewhere in the statistical literature (for example, on page 22). Readers should bear in mind the distinguished, although disrupted, career of the author and make proper allowance for certain peculiarities of writing which are not usual in the English language. Examples are, "initial median" for the median of the parent population, and "analysis statistics" for methods of statistical analysis - as well as the extensive use of the term "exceedance".

The eight chapters: Aims and tools; Order statistics and their exceedances; Exact distribution of extremes; Analytical study of extremes; The first asymptotic distribution; Uses of the first asymptote; The second and third asymptotes; The Range are freely sub-divided and contain summary sections from time-to-time. Of special practical interest are sections 6.3 and 7.3 which deal with typical applications of extreme-value theory. There are, of course, minor slips and printer's errors to be found but the book of this kind has yet to be produced that has not its shares of these troubles. What is important is that they are nowhere misleading and to add to the great value of the subject-matter we have revealing glimpses of the pioneer spirit of the author (for example, on page 6 in connection with methods of estimation). Finally, this book is another example of that rather engaging feature, made famous by a well-known economist, of heading some chapters and sections with a well chosen thought or quotation. This reviewer especially likes the one on page 238 which reads,

Half a loaf is better than none
(Half a bridge ....................)

In spite of its high price, this book is a must for any practising statistician who has to deal with problems of extreme values.

8.9. Review by: Alan Stuart.
Economica, New Series 27 (106) (1960), 197.

It is still widely believed that statistical theory must always be essentially concerned with the behaviour of averages or similar composite quantities, that the statistician cannot cope with the exceptional, extreme members of a group. This book is entirely devoted to the statistical theory of "extreme" observations, which are the observations at or close to the largest and smallest in a sample. Extreme-value theory has considerable mathematical fascination, and contains some notable results, the most striking of which is worth mentioning here. As has been known for 30 years, no matter what the shape of the initial distribution from which we take a large sample, the distribution of the largest (or of the smallest) value observed must be one of three certain distributions; which of these three distributions holds in any particular case depends only on the general shape of the "tail" of the initial distribution. A good deal of this book is concerned with the detailed study of these three distributions and the relationships between them.

Apart from its mathematical interest, extreme-value theory has been successfully applied in many fields; indeed, the practical needs have often evoked the theory. Especially successful applications have been to the strength of structures and materials subject to occasional extreme stresses, and thus to problems of flood control; and to meteorological observations of many kinds, including problems of drought control and thence, by implication, to dam construction specifications. There is some discussion of the applications in this book and Professor Gumbel, who has devoted most of his working life to the theory and applications of extreme-values, is concerned to underline the practical lessons of the theory he derives (and occasionally even lovingly re-derives). But despite this and despite the extensive use of graphical and tabular presentation of theoretical results, this is essentially a book for the professional statistician. The less specialised reader would profit more from the same author's earlier pamphlet Statistical Theory of Extreme Values and Some Practical Applications (published in 1954 as No. 33 in the U.S. Government's National Bureau of Standards, Applied Mathematics Series) and even the intending specialist new to this field might take this pamphlet as a preliminary dose. The book contains an excellent and highly international bibliography, and an index quite inadequate for what is at present the only, and for many years will surely be the leading, reference book in this field.