**Lehrbuch der Algebra (1895-96), by Heinrich Weber.****Review by: James Pierpont.**

*Bull. Amer. Math. Soc.***4**(5) (1898), 200-234.For some years the need of a thoroughly modern textbook on algebra has been seriously felt. The great strides that algebra has taken during the last twenty-five years, in almost all directions, have made Serret's classical work more and more antiquated, while modern books like Petersen's and Carnoy's make no claims to give a large and comprehensive survey of the subject. The appearance of any book modelled on the same broad plan as Serret's 'Algèbre Supérieure' would thus be greeted with a hearty welcome, but when written by such a master as Heinrich Weber, we greet it with expressions of sincerest joy and satisfaction. As Weber himself tells us, he has cherished for years the plan of this great undertaking ; but before deciding to execute it he has traversed in his university lectures many times this vast domain as a whole, and has treated various parts separately with greater detail. No wonder, then, that we find everywhere in perusing the pages of his work, evidence of mature judgment in the choice of the matter treated, in the amount of space devoted to each subject according to its relative importance, in the harmonious grouping of the parts, in the manner of presenting the different theories. Our admiration is no less excited by its pedagogic excellencies ; Weber's German is simple and concise, the demonstrations are clear and rigorous, and many of them are of extreme elegance. After a point of view has been gained, the author disposes of one problem after the other as they are met in his path, and, as these are always clearly stated, the reader has the pleasure of knowing in advance the goal in view. ... Now in regard to the plan of the work: Weber's idea has been to write a work which shall lead the reader by easy steps from the most elementary notions to the most advanced and modern theories. Various auxiliary notions and theories are developed with great skill as found necessary, so that the reader nowhere needs to consult other works for information in order to continue his reading with pleasure and profit. As was to be expected, the chief interest of the work centres in the distinctly modern theories of Galois and Klein, of finite groups and algebraic numbers, while the older invariant theory as developed in the Higher Algebra of Salmon, for example, is scarcely touched. ... we wish to note here that, in our opinion, the author's presentation of Galois's theory, with its many excellent examples, is beyond all comparison the most satisfactory yet given, whether in a work on algebra or one devoted exclusively to this theory; we recommend it most heartily for all who wish to acquaint themselves with this difficult but essential part of modern algebra. No less heartily can we praise the section devoted to explain Kronecker's theories applied to algebraic numbers. This certainly is the most novel part of the whole work, and Weber has earned by it the thanks of the whole mathematical community. We do not doubt that many will find Kronecker's algebraic methods more congenial than the purely arithmetical but more abstract methods of Dedekind. ... A classic from the day of its publication, it is destined to a long and useful career, a monument of honour to its genial author.

**Lehrbuch der Algebra (2nd edition) (1898-99), by Heinrich Weber.****Review by: James Pierpont.**

*Bull. Amer. Math. Soc.***5**(10) (1899), 480-482.Most of the readers of the Bulletin have been aware that a new edition of Weber's Algebra was in progress. Some time ago the first volume appeared ; the work is now complete, the second volume of the new edition having just come out. The many excellencies of this great work have been so generally appreciated that Professor Weber has experienced the very unusual pleasure of seeing a new edition required in less than two years after the publication of the first. ... We rejoice to learn, by remarks made in two places, that the author still contemplates writing a third volume, which is destined to treat of the application of the general theory of algebraic numbers to the complex multiplication of elliptic functions. We are sure we express the unanimous sentiment of the whole mathematical community in hoping that Professor Weber will resolutely persevere in this arduous undertaking.

**Die Partiellen Differentialgleichungen der Mathematischen Physik (4th edition), Vol 1 (1900), by Heinrich Weber.****Review by: G B Mathews.**

*The Mathematical Gazette***2**(27) (1901), 51-52.This differs so essentially from the last edition as to be practically a new work. But Professor Weber is fully justified in keeping Riemann's name on the title-page, not only as a pious memorial, but also because he has preserved, as far as possible, the substance of Riemann's work, and in making the requisite additions has happily succeeded in maintaining the spirit of his great predecessor. Among the principal new features in this volume may be mentioned the sections on Bessel functions, vectors, and spherical harmonics, together with a whole book (pp. 305-506) on electricity and magnetism. The analysis is extremely elegant, and shows inter alia the advantage of a judicious use of the vector notation. ... The order of arrangement of subjects may be thought rather curious; certainly hydrodynamics and the conduction of heat present less difficulty to a student than electricity does, and naturally precede it in a course of lectures. But Professor Weber is not writing for the average student who is beginning the study of physics, and his arrangement is possibly suggested by a consideration of the analysis required

**Die Partiellen Differentialgleichungen der Mathematischen Physik (4th edition), Vol 1 (1900) and Vol 2 (1901), by Heinrich Weber.****Review by: Joseph Sweetmen Ames.**

*Bull. Amer. Math. Soc.***8**(2) (1901), 81-85.Riemann's lectures on the partial differential equations of mathematical physics and their application to heat conduction, elasticity, and hydrodynamics were published after his death by his former student, Hattendorff. Three editions appeared, the last in 1882; and few books have proved so useful to the student of theoretical physics. The object of Riemann's lectures was twofold : first, to formulate the differential equations which are based on the results of physical experiments or hypotheses ; second, to integrate these equations and explain their limitations and their application to special cases. However, since Riemann's death, the development and extension of physics has been so great, especially in our knowledge of the properties of the luminiferous ether and of the properties of moving electrical charges - ions, and the advances in mathematical analysis have been so marked, that simply a revision of Hattendorff's treatise has become impossible. It has been felt for some years that what was most needed by students of physics was not a new edition of Riemann's lectures, but rather an entirely new book with the same purposes in view and with possibly the same general title. This has been given us at last by Professor Heinrich Weber, of Strasburg. The fact that Professor Weber still retains on the title-page of his two volumes the words, "nach Riemann's Vorlesungen" is certainly a tribute of the highest respect of one great scholar for another; for to Weber, and to him alone, belongs all the credit for this most important work. These two volumes include all the subjects - and more - treated by Riemann in the two books which were prepared for publication by Hattendorff ... If it is possible to select from the many most excellent chapters of Professor Weber's book that one which seems the most noteworthy, that chosen would be the one entitled a "Brief discussion of the principles of mechanics." In this we have, in the short space of thirty-three pages, the most satisfactory deduction and discussion of Hamilton's and Lagrange's equations that is known to the reviewer. ... there is no other book in any language which provides the student of physics with so much necessary information, so well selected and prepared.

**Encyklopädie der Elementar-Mathematik, Vol 1 (1903), by Heinrich Weber and Josef Wellstein.****Review by: David Eugene Smith.**

*Bull. Amer. Math. Soc.***10**(4) (1904), 200-204.It was with much interest that scholars learned, a few years ago, of the proposed appearance of an Encyklopädie der Mathematischen Wissenschaften, and this under circumstances to assure the highest standard of excellence. The publication of this monumental work was begun in 1898, and the parts which have thus far appeared have in general fulfilled the expectations of mathematicians. It has been international in its view and in its contributions, rich in its bibliography, and comprehensive in scope. It has, however, failed to touch the field of elementary mathematics in such a way as to be helpful to the great body of secondary teachers and students. Recognizing the need for a more elementary work, the same publishers have undertaken the Encyklopädie der Elementar-Mathematik, originally planned for two volumes, but now announced to include a third on the applications of the subject. The general scheme is quite as commendable as that of the more pretentious work in the higher fields, for students and teachers in elementary lines are even more in need of helpful suggestions as to bibliography, modern methods, and the improvement of their subjects from the standpoint of recent mathematics, than are the investigators in the advanced theory. A difficulty presents itself, however, in defining the term "Elementary mathematics," and here the authors confessedly lay themselves open to criticism. They admit that all attempts at fixing the limits on purely mathematical grounds have been unsatisfactory, and they have been compelled to resort to educational considerations instead. It must be confessed that the result has not been altogether satisfactory, and it is difficult to reconcile it with the announced position taken by the authors. ... the work is arranged and written rather in the style of a higher algebra than an encyclopaedia in the English or French sense. It is certainly in no respect a

*Lexihon,*even with its excellent index, and on the whole, by those who looked for a book of general information on elementary mathematics, it will be considered disappointing. But aside from the technical use of the name, which is quite correct from the Teutonic standpoint, one still has good cause for disappointment. He looks for a general and exhaustive treatise, and finds only a somewhat condensed higher text-book ; he looks for a wide range of topics, and finds only a partial list of even the essentials ; he looks for help in the way of bibliography, but he finds only a few commonplaces .... A teacher of the elements might reasonably expect, for example, to find some systematic treatment of factors in an encyclopaedia of mathematics ; he might hope for some suggestions on the subjects of graphs or undetermined coefficients ; and he might feel that here he could have light thrown upon some of the vexing questions of extraneous roots, radical equations, and the exact meaning to be attached to certain symbols. He might also hope for some adequate treatment of elementary determinants, possibly a few suggestions as to the theory of numbers, and very likely a brief treatment of a topic like numerical computations. None of these topics will he find discussed, however, and some are not even mentioned. I t might also be expected that some efforts would be made to supply historical notes of interest to one who is teaching or reviewing the subject, even if not of value to one engaged in research. Such notes as appear are, however, practically worthless. ... With all due appreciation of the scholarship of the work, and of its helpfulness, it must therefore be a matter of regret to all who have looked forward to its appearance, that the ground covered is not that of elementary mathematics in an international sense, that the historical notes are very ill considered, that no attempt has been made to offer a helpful bibliography, and that the arrangement and general treatment are so far removed from that of the Repertorium or the Burkhardt-Meyer Encyklopädie.**Encyklopädie der Elementar-Mathematik, Vol 2 (1905) and Vol 3 (1907), by Heinrich Weber and Josef Wellstein.****Review by: Henry Seely White.**

*Bull. Amer. Math. Soc.***14**(10) (1908), 499-501.Every live teacher of secondary school mathematics is aware of the superficial character of most text-books. In the nature of the case, teaching for younger pupils must exclude far more than it presents. But if the teacher himself restricts his study to the range prescribed for pupils, very little mathematical interest is kindled in his classes. Practically the same result is reached if the sole scientific interest of the teacher is in fields remote from his pupils' studies. The authors of this three-volume encyclopaedia of elementary mathematics plan to intensify by fundamental criticism, and revivify by extensive applications to questions of physics, the interest of the young teacher in his every-day work. This does not conflict by any means with the programme of modern universities - to train the future teacher by research in some region on the frontiers of scientific knowledge. Rather it supplements that programme, and strengthens the position of its champions, by showing how to apply the method of the university seminar to the problems of the schoolroom. ... In a note appended to this [third] volume, H Weber reverts to the Mengenlehre of the first volume, cites Russell's paradox on the class of classes that do not contain themselves (which he identifies with one of Kant's antinomies); and gives an outline discussion of finite aggregates, free from objections, as he believes. These volumes certainly constitute a valuable work for every reference library.

**Encyklopädie der Elementar-Mathematik, Angewandte Elementar-Mathematik Vol 3 (1907), by Heinrich Weber and Josef Wellstein.****Review by: Editors.**

*The Mathematical Gazette***4**(67) (1907), 163.This third volume of the Encyclopaedia of Elementary Mathematics is devoted to applied mathematics. As the authors explain in the preface, this volume is not encyclopaedic in the ordinary sense, for it makes no pretence to a complete treatment of the subjects dealt with. It is intended rather as a careful exposition of the principles of those branches of applied mathematics from which school courses are compiled. The interest of the book to an English teacher lies chiefly in the selection and relative proportion of the topics dealt with, namely, Vector Geometry, Analytical Statics and Dynamics, Electric and Magnetic Lines of Force, Geometric Maxima and Minima, Probability and Theory of Errors, Descriptive Geometry, Graphic Statics.

**Encyklopädie der Elementar-Mathematik. Vol 1. Elementare Algebra und Analysis (3rd edition) (1909), by Heinrich Weber and Josef Wellstein.****Review by: Frederick William Owens.**

*Bull. Amer. Math. Soc.***17**(10) (1911), 546.The third edition differs from the first primarily in the introduction of nearly 100 pages of new subject matter. These additions consist chiefly of historical notes, which are appended to a number of chapters, and an entirely new chapter at the end, which is devoted to the graphical representation of a function, differentiation, and integration. The value of the book is much increased by the additions, which harmonize with the spirit of the earlier editions.

**Die Partiellen Differentialgleichungen der Mathematischen Physik (5th edition), Vol 1 (1910), by Heinrich Weber.****Review by: Oliver Dimon Kellog.**

*Bull. Amer. Math. Soc.***18**(2) (1911), 87-89.The earlier editions of this work have won for themselves an important place in every mathematical library, where they are frequently consulted both as reference books and for more consecutive reading supplementary to courses on mathematical physics. They have thus become so well known that an extensive review of the volume before us would be superfluous. ... In the preface to the fifth edition, Professor Weber remarks that some of the developments of mathematical physics during the past decade have been too significant to leave unmentioned, and that although they have not been carried sufficiently far to permit of finished exposition in a text-book, they should receive some attention in the pages to follow. He speaks of integral equations and the notion of relativity as two of the more important contributions made to the science since his last edition in 1900, and promises an application of the first in the present volume. The failure to find any satisfactory fulfilment of this promise will be the reader's greatest disappointment. ... In general the author has been conservative in making changes. More valuable additions might have been made in some cases. Linear partial differential equations of second order, for instance, receive a scant two pages, which give only the device of finding particular solutions in the form of exponential functions with linear exponents. Undoubtedly the systematic development of integral equations and their applications, of the principle of relativity, and in all likelihood, of Lebesgue's theory of integration, will make necessary at some future time a book treating the same subjects in radically new ways. But for the present, and for some time to come, Weber's book, with its origins in Riemann's lectures, will continue to be indispensable. The new edition will find its way into the more complete libraries, although for smaller collections the previous one will prove well-nigh as useful, if a prediction can be made on the basis of the first volume of the new edition.

**Encyklopädie der Elementar-Mathematik. Angewandte Elementar-Mathematik. (1st part, Vol 3) (2nd edition) (1910), by Heinrich Weber and Josef Wellstein.****Review by: James Byrnie Shaw.**

*Bull. Amer. Math. Soc.***19**(2) (1912), 87-88.In this second edition of the third volume, on applied mathematics, there are extensive changes. The original volume is divided into two; the present one, a treatise complete in itself on mathematical physics, and one to follow on graphics, probabilities, and astronomy. This modification has been made to satisfy many criticisms of the original, some of which deplored the wide omissions in a work that called itself an encyclopedia. The present volume has three chapters on mechanics: functions of position and direction that appear in physics, analytic statics, and dynamics; two chapters on electric and magnetic fields: electricity and magnetism, and electromagnetism; two chapters on maxima and minima: geometric maxima and minima, and applications to the theories of equilibrium and of capillarity; two chapters on optics: geometric optics, and plane waves.

**Encyklopädie der Elementar-Mathematik. Angewandte Elementar-Mathematik. (2nd part, Vol 3) (2nd edition) (1912), by Heinrich Weber and Josef Wellstein.****Review by: James Byrnie Shaw.**

*Bull. Amer. Math. Soc.***19**(8) (1913), 422-423.This second edition of the Elementary Encyclopaedia has received such extensive additions that the third volume of the original appears in two parts. ... The second part, under consideration, contains the revised books entitled "Graphik" and "Wahrscheinlichkeitsrechnung." The first book has a new section on "Axonometrie und Perspektive." Two new books have been added to meet the views of certain critics of the first edition: "Politische Arithmetik" and "Astronomic" Other changes are minor. The third book includes the theory of interest and actuarial computations. The theory of interest is based upon compound interest, in the sense that simple interest is looked upon as an annuity in perpetuity. Only the elements of insurance are developed. The fourth book deals with spherical astronomy and the calculation of orbits. The subjects considered are astronomical coordinates, determination of time, variations of stellar coordinates, observations with instruments, determination of latitude and longitude, and orbits. The additions to this useful work will be welcome in many quarters. While one might criticize the proportional amount of space devoted to them, and to the other divisions of the book, such criticism would arise from purely personal views as to what applications are important, and would vary from person to person. The authors and editors are deserving of praise for the work taken as a whole.

**Lehrbuch der Algebra. Kleine Ausgabe in einem Bande (1912), by Heinrich Weber.****Review by: G A Miller.**

*Science, New Series***38**(981) (1913), 550-551.Among the advanced text-books on algebra there is probably none which is more favourably known than Weber's "Lehrbuch der Algebra" in three large volumes. The great extent of the work doubtless discouraged many beginners as well as those who have only time to learn the fundamental principles of this vast subject. Hence the small volume before us should find a hearty welcome among many students of mathematics who understand the German language. ... Although most students who are in position to profit much by the study of such a work can read German, yet there is doubtless a considerable number to whom an English translation would be very helpful, since there is no algebra in the English language which covers the same ground. ... In the preface it is stated that the author was assisted by his colleagues, especially by Messrs. Löwy, Epstein and Levi, while correcting the proof. These names, together with that of H Weber, are a sufficient guarantee that no important errors appear in the book.

**Encyklopädie der Elementar-Mathematik. Vol 1. Arithmetik, Algebra und Analysis (4th edition) (1922), by Heinrich Weber, Josef Wellstein and Paul Epstein.****Review by: J W Young.**

*Bull. Amer. Math. Soc.***29**(10) (1923), 480.Since the publication of the [third edition], both the original authors have died. The new (fourth) edition of the first volume appears under the editorship of Paul Epstein. ... The most noteworthy is the omission of the short chapter on differentiation, which was introduced first in the second edition, and which is now again abandoned partly to save space for material that appears to the present editor more important and partly because a mere brief introduction is not in keeping with plan of the work. ... The present edition is in spite of the omission noted somewhat larger than the previous one, owing to the inclusion of some new material and the elaboration of some of the older material. ... The historical and bibliographical notes seem to have been distinctly improved. There can be no doubt that the new edition will adequately fill the place in our standard reference literature which the earlier editions filled with such marked success.

**Encyklopädie der Elementar-Mathematik. Vol 1. Arithmetik, Algebra und Analysis (5th edition) (1934), by Heinrich Weber, Josef Wellstein and Paul Epstein.****Review by: TAAB.**

*The Mathematical Gazette***19**(233) (1935), 159-160.The reader unfamiliar with this book should perhaps be told that the word Encyclopaedia is used in a somewhat modified sense. The book is not a classified collection of all results which may be said to be elementary, nor is it an enumeration of the methods of elementary mathematics. It is an attempt, based on a really prodigious amount of labour carried out with a truly Teutonic thoroughness, to give the reader an opportunity of relating any really important branch of elementary mathematics - that is, roughly speaking, mathematics up to the beginnings of the calculus - to the logical foundations as well as to the appropriate non-elementary extensions of that branch. Thus, for example, the complex number has its elementary geometrical significance fully expounded, while the formal algebraic development, starting from the definition of a complex number as an ordered pair of real numbers, is made to lead naturally towards the investigation of quaternion and vector algebras where the commutative law breaks down. ... His work has been by no means superficial, for the table of contents is marked by asterisks which show that of the 139 paragraphs now contained in the book, more than half are either new or completely rewritten versions of the original. Not only have methods once fashionable been superseded, but results which were in 1903 neither elementary nor widely appreciated, are in 1935 perhaps still not elementary but have so permeated the whole of mathematics that their impact and influence on elementary work is considerable.