## Reviews of the *Enciclopedia delle Matematiche Elementari*

Below we present reviews of the first two volumes (in four parts) of the

*Enciclopedia delle Matematiche Elementari*1.

**G A Miller, Review: Enciclopedia delle Matematiche Elementari**(Vol 1, Part 1), (Vol 1, Part 2), by L Berzolari, G Vivanti and D Gigli,

*Bull. Amer. Math. Soc.*

**38**(3) (1932), 157-158.

The preparation of an Italian encyclopaedia of elementary mathematics was approved by the Society

The twenty main headings of the present volume, with the number of pages devoted to each of these subjects and the authors of the articles, are as follows: Logic, (75), A Padoa; General arithmetic, (126), D Gigli; Practical arithmetic, (52), E Bortolotti and D Gigli; Theory of numbers and indeterminate analysis, (68), M Cipolla; Progressions, (17), A Finzi; Logarithms, (42), A Finzi; Mechanical calculus, (28), G Tacchella; Combinatory calculus, (9), L Berzolari; Elements of the theory of groups, (51), L Berzolari; Determinants, (30), L Berzolari; Linear equations, (13), L Berzolari; Linear substitutions and linear, bilinear and quadratic forms, (28), L Berzolari; Rational functions of one or more variables, (37), O Nicoletti; General properties of algebraic equations, (59), O Nicoletti; Equations of the second, third, and fourth degree, and other particular algebraic equations, systems of algebraic equations of elementary type, (57), E G Togliatti; Methods for the discussion of problems of the second degree and remarks on some of the third and fourth degree, (63), R Marcolongo, Limits, series, continued fractions, and infinite products, (45), G Vitali; Elements of infinitesimal analysis, (101), G Vivanti; Relations between the theory of aggregates and elementary mathematics, (11), G Vivanti; The analytic function from an elementary point of view, (29), S Pincherle.

Each of the two parts of this volume closes with an author index, but the volume contains no subject index. It contains an unusually large number of references to sources and hence it is very useful to those interested in the history of elementary mathematics. The large amount of space devoted to such subjects as infinitesimal analysis and the theory of groups exhibits the fact that modern developments receive considerable attention and that the authors do not share the view that the developments of elementary mathematics were practically completed at the close of the seventeenth century. On the contrary, they point out the well known fact that many of the developments in modern advanced mathematics extend into the elements of our subject and throw new and useful light on these elements. Fifty years ago not many mathematicians would have been so bold as to predict that in 1931 an encyclopaedia of elementary mathematics would devote more than fifty pages to algebraic group theory, as is done in the present work.

This work differs widely from the only other large modern encyclopaedia of elementary mathematics, by Weber and Wellstein. It is much less in the form of a treatise on the subjects concerned, but it aims to prove the most fundamental theorems and to give abundant references to places where the proofs of others can be found, especially the earliest ones. By limiting itself to the elements, the exposition becomes comparatively brief, and this enables the reader to gather here with comparative ease the main facts relating to a wide range of subjects. The subject which is treated at the greatest length is general arithmetic, and the subject to which the least number of pages is devoted is combinatory calculus. The work is especially useful to the younger students of our subject who desire to become acquainted with the main elements of the various fields of mathematics before entering deeply into any one of them. It should also be very useful to teachers of secondary mathematics who can thereby find a large amount of modern material and of historical data relating to the subjects which they are teaching.

Near the beginning of each of the two parts of the present volume there appears a list of mathematical publications with the abbreviations therefor used in the body of this work. The second part contains 155 such publications but only 10 of these are American. The most conspicuous of the missing ones is the Proceedings of the National Academy of Sciences, which now contains a large number of mathematical articles. On page 410 and in the index of Part 1 there appears the name F R Moulton instead of J F Moulton, and on page 19 of Part 2, the date of Ruffini should be 1799 in place of 1798. On page 43 of this part it is stated that the natural numbers form an infinite abelian group when they are combined by multiplication. This error was, however, corrected, even before it was made, on page 33, note 77. The same error appears also in the first edition of volume 2 of Weber's 'Lehrbuch der Algebra', as well as in many other places. On page 314 and in the index the name J Kempner should be replaced by A J Kempner. The number of such oversights is fortunately relatively small, so that the volume as a whole can be heartily recommended. It would be very desirable to have a similar work in English notwithstanding the fact that a considerable portion of the modern mathematical language is universal and hence English speaking students of mathematics can use this work without much language trouble.

2. *Mathesis*as early as 1909, and the project was then viewed with favour also in other countries so that it seemed desirable to publish a German edition contemporaneously with the original. Active preparation was begun at once and in 1916 the proposed publication was announced in my*Historical Introduction to Mathematical Literature*, page 279, as well as in other places. Various obstacles, including the World War, delayed the publication so that the first volume, which is devoted to analysis, was only recently completed after being brought up to date by the authors of the articles contained therein. The remaining two volumes are to treat geometry and applied mathematics respectively. The latter volume is expected to include also a discussion of the teaching and the history of mathematics.The twenty main headings of the present volume, with the number of pages devoted to each of these subjects and the authors of the articles, are as follows: Logic, (75), A Padoa; General arithmetic, (126), D Gigli; Practical arithmetic, (52), E Bortolotti and D Gigli; Theory of numbers and indeterminate analysis, (68), M Cipolla; Progressions, (17), A Finzi; Logarithms, (42), A Finzi; Mechanical calculus, (28), G Tacchella; Combinatory calculus, (9), L Berzolari; Elements of the theory of groups, (51), L Berzolari; Determinants, (30), L Berzolari; Linear equations, (13), L Berzolari; Linear substitutions and linear, bilinear and quadratic forms, (28), L Berzolari; Rational functions of one or more variables, (37), O Nicoletti; General properties of algebraic equations, (59), O Nicoletti; Equations of the second, third, and fourth degree, and other particular algebraic equations, systems of algebraic equations of elementary type, (57), E G Togliatti; Methods for the discussion of problems of the second degree and remarks on some of the third and fourth degree, (63), R Marcolongo, Limits, series, continued fractions, and infinite products, (45), G Vitali; Elements of infinitesimal analysis, (101), G Vivanti; Relations between the theory of aggregates and elementary mathematics, (11), G Vivanti; The analytic function from an elementary point of view, (29), S Pincherle.

Each of the two parts of this volume closes with an author index, but the volume contains no subject index. It contains an unusually large number of references to sources and hence it is very useful to those interested in the history of elementary mathematics. The large amount of space devoted to such subjects as infinitesimal analysis and the theory of groups exhibits the fact that modern developments receive considerable attention and that the authors do not share the view that the developments of elementary mathematics were practically completed at the close of the seventeenth century. On the contrary, they point out the well known fact that many of the developments in modern advanced mathematics extend into the elements of our subject and throw new and useful light on these elements. Fifty years ago not many mathematicians would have been so bold as to predict that in 1931 an encyclopaedia of elementary mathematics would devote more than fifty pages to algebraic group theory, as is done in the present work.

This work differs widely from the only other large modern encyclopaedia of elementary mathematics, by Weber and Wellstein. It is much less in the form of a treatise on the subjects concerned, but it aims to prove the most fundamental theorems and to give abundant references to places where the proofs of others can be found, especially the earliest ones. By limiting itself to the elements, the exposition becomes comparatively brief, and this enables the reader to gather here with comparative ease the main facts relating to a wide range of subjects. The subject which is treated at the greatest length is general arithmetic, and the subject to which the least number of pages is devoted is combinatory calculus. The work is especially useful to the younger students of our subject who desire to become acquainted with the main elements of the various fields of mathematics before entering deeply into any one of them. It should also be very useful to teachers of secondary mathematics who can thereby find a large amount of modern material and of historical data relating to the subjects which they are teaching.

Near the beginning of each of the two parts of the present volume there appears a list of mathematical publications with the abbreviations therefor used in the body of this work. The second part contains 155 such publications but only 10 of these are American. The most conspicuous of the missing ones is the Proceedings of the National Academy of Sciences, which now contains a large number of mathematical articles. On page 410 and in the index of Part 1 there appears the name F R Moulton instead of J F Moulton, and on page 19 of Part 2, the date of Ruffini should be 1799 in place of 1798. On page 43 of this part it is stated that the natural numbers form an infinite abelian group when they are combined by multiplication. This error was, however, corrected, even before it was made, on page 33, note 77. The same error appears also in the first edition of volume 2 of Weber's 'Lehrbuch der Algebra', as well as in many other places. On page 314 and in the index the name J Kempner should be replaced by A J Kempner. The number of such oversights is fortunately relatively small, so that the volume as a whole can be heartily recommended. It would be very desirable to have a similar work in English notwithstanding the fact that a considerable portion of the modern mathematical language is universal and hence English speaking students of mathematics can use this work without much language trouble.

**A A Bennett, Review: Enciclopedia delle Matematiche Elementari**(Vol 1, Part 1), by L Berzolari, G Vivanti and D Gigli,

*Amer. Math. Monthly*

**37**(7) (1930), 378-380.

The present "Part 1", is to be followed at once by "Part 2," announced as already under press. Volume 1 is devoted to analysis, and is to be one of three volumes constituting this encyclopedia. The second volume is to deal with geometry and the third with applications of mathematics, history of mathematics and didactic questions. The present part comprises seven independent articles together with indices etc., namely : I. Logic, by Alessandro Padoa; II. General arithmetic, by Duilio Gigli; Il l. Practical arithmetic, by Ettore Bortolotti and Duilio Gigli; IV. Theory of numbers and indeterminate analysis, by Michele Cipolla; V. Progressions, by Aldo Finzi; VI. Logarithms, by Aldo Finzi; VII. Mechanical calculation, by Giuseppe Tacchella.

This is the first work of this sort in the Italian language and the editorial commission has approached its task with an enthusiasm that suggests not merely service in a needed field, but also patriotic fervor, as seen throughout the preface dated "1929 (A. VII )." The form and subject matter may well be contrasted with that of Pascal's "Repertorium" (not mentioned in this work) whose expanded German editions are proving to be of the highest service for ready reference. The treatment here used is intended to meet the demand of teachers and younger students who do not have ready access to basic sources and the material to be covered is intended to be roughly that touched by the end of the first two years of university courses.

Any elaborately detailed and critical analysis of each of the seven condensed treatises which are here offered to the reader will be inappropriate. The American teacher will find most of the material elsewhere in a language more familiar to him. Despite excellent historical references, completeness in this book is not promised . The value of the work cannot consist in its completeness, nor in its bringing to light unsuspected historical sources. It must be judged by the degree to which it renders available to college and secondary school teachers the important elements touching upon the basic mathematical courses, and the good judgment with which topics from so vast a possible field are selected and woven in to smoothly flowing expositions. The authors have refrained from that hint of superiority that might be suggested by the familiar phrase, "vom hoheren Standpunkte aus" and incidentally have missed much of the lively interest for the graduate student that characterizes Klein's work of a somewhat similar purpose (

The first section (of 79 pages) is a serious study of the formal elementary aspects of modern logic as used by Peano and later by Russell and others. This section is not referred to in later parts and would seem to be somewhat out of keeping with the general tenor of the work, despite the fact that mathematics uses logic and to a considerable extent is logic. Controversial questions and all references to paradoxes are suppressed, but the reader may agree with the reviewer in feeling that the theories have not yet been subjected to that intensive critical analysis by many independent workers which is the only practicable test of scientific stability. The astonishing number of 143 independent (?) axioms are listed consecutively.

The section on general arithmetic (covering 130 pages) deals with what is often called the number-concept, starting with finite classes and natural numbers and concluding with the transcendence of

The third section devoted to "practical arithmetic" is what many persons might call the abstract theory of arithmetic. Apart from some interesting historical topics the section deals in a logical but thoroughly elementary way with the fundamental operations of arithmetic, and the proofs incident thereto (assuming much prior logical foundation). Questions concerning mensuration, percentage and other topics of commercial arithmetic find no place. The fourth section treats of certain topics in elementary number theory. Apart from a very brief preliminary discussion, two main divisions deal respectively with linear congruences (Fermat-Euler theorem, the Gaussian, Wilson's theorem, Euclid's algorithm, etc.), and with higher congruences (roots of unity, quadratic congruences, etc.). In no other section does one find so many familiar topics omitted, but what has been retained shows an unusual attempt at continuity. The treatment is simple, interesting and in connection with topics treated reasonably comprehensive.

The fifth section, dealing with progressions, seems in strange company. It discusses only the traditional arithmetic, geometric, and harmonic series and some of the most elementary extensions of the first two to figurate numbers. The use of

3. This is the first work of this sort in the Italian language and the editorial commission has approached its task with an enthusiasm that suggests not merely service in a needed field, but also patriotic fervor, as seen throughout the preface dated "1929 (A. VII )." The form and subject matter may well be contrasted with that of Pascal's "Repertorium" (not mentioned in this work) whose expanded German editions are proving to be of the highest service for ready reference. The treatment here used is intended to meet the demand of teachers and younger students who do not have ready access to basic sources and the material to be covered is intended to be roughly that touched by the end of the first two years of university courses.

Any elaborately detailed and critical analysis of each of the seven condensed treatises which are here offered to the reader will be inappropriate. The American teacher will find most of the material elsewhere in a language more familiar to him. Despite excellent historical references, completeness in this book is not promised . The value of the work cannot consist in its completeness, nor in its bringing to light unsuspected historical sources. It must be judged by the degree to which it renders available to college and secondary school teachers the important elements touching upon the basic mathematical courses, and the good judgment with which topics from so vast a possible field are selected and woven in to smoothly flowing expositions. The authors have refrained from that hint of superiority that might be suggested by the familiar phrase, "vom hoheren Standpunkte aus" and incidentally have missed much of the lively interest for the graduate student that characterizes Klein's work of a somewhat similar purpose (

*Elementarmathematik vom hoheren Standpunkte aus,*Third Edition, 1924-1 928). One ought rather to compare this work with the H Weber and J Wellstein,*Encyklopädie der Elementar-Mathematik*(Vol. 1, 4th Edition 1922, Vol. 2, 2nd Edition 1907. Vol. 3, 1st Edition 1907).The first section (of 79 pages) is a serious study of the formal elementary aspects of modern logic as used by Peano and later by Russell and others. This section is not referred to in later parts and would seem to be somewhat out of keeping with the general tenor of the work, despite the fact that mathematics uses logic and to a considerable extent is logic. Controversial questions and all references to paradoxes are suppressed, but the reader may agree with the reviewer in feeling that the theories have not yet been subjected to that intensive critical analysis by many independent workers which is the only practicable test of scientific stability. The astonishing number of 143 independent (?) axioms are listed consecutively.

The section on general arithmetic (covering 130 pages) deals with what is often called the number-concept, starting with finite classes and natural numbers and concluding with the transcendence of

*e*and π. The treatment is excellent although necessarily far from novel. It may be too abstract to suit many college teachers in this country, but would be worth any effort required to understand it.The third section devoted to "practical arithmetic" is what many persons might call the abstract theory of arithmetic. Apart from some interesting historical topics the section deals in a logical but thoroughly elementary way with the fundamental operations of arithmetic, and the proofs incident thereto (assuming much prior logical foundation). Questions concerning mensuration, percentage and other topics of commercial arithmetic find no place. The fourth section treats of certain topics in elementary number theory. Apart from a very brief preliminary discussion, two main divisions deal respectively with linear congruences (Fermat-Euler theorem, the Gaussian, Wilson's theorem, Euclid's algorithm, etc.), and with higher congruences (roots of unity, quadratic congruences, etc.). In no other section does one find so many familiar topics omitted, but what has been retained shows an unusual attempt at continuity. The treatment is simple, interesting and in connection with topics treated reasonably comprehensive.

The fifth section, dealing with progressions, seems in strange company. It discusses only the traditional arithmetic, geometric, and harmonic series and some of the most elementary extensions of the first two to figurate numbers. The use of

**x****y**to denote**x**∧**y**is the most noticeable single item of interest. The section does, of course, form a natural bridge from linear congruences to the introduction of logarithms. The sixth section, on Logarithms, is more than usually effective and covers material ordinarily assumed to be too detailed for elementary texts and too special for advanced mathematical courses. The seventh section, on computing machines, suffers only from the common difficulty incidental to merely talking about mechanical devices whose virtues lie in their concrete utility. While one machine actually used would outweigh in significance any number of mere descriptions the reader is at least prepared to admit that there are many machines and for divers purposes. The paper is poor, the binding flimsy, the figures cheaply drawn, and typographical mistakes not unknown (The table of contents lists "Errata, pag. 415." It should read "P. 451."). But throughout the presentation is scholarly, the emphasis welt-placed , and the language simple, connected and interesting. Whether the completed work will prove eventually more attractive to American readers than the somewhat more conventional "Weber-Wellstein," will remain to be seen.**G A Miller, Review: Enciclopedia delle Matematiche Elementari**(Vol 1, Part 2), by L Berzolari, G Vivanti and D Gigli,

*Amer. Math. Monthly*

**39**(3) (1932), 168-171.

The fact that there is no modern mathematical encyclopaedia in the English language is perhaps sufficient evidence of the magnitude of such an undertaking even if it is restricted to the elements of our subject. Part I of the present volume appeared in 1930 and was reviewed in this MONTHLY, volume 37 (1930), page 378, by Professor A A Bennett. The second and somewhat larger part of this volume is dated 1932 and completes the treatment of the subject of analysis in a little over a thousand pages, while the

The thirteen main headings of this second part, with the number of pages devoted to each of these subjects and the authors of the articles, are as follows : combinatory calculus (9), L Berzolari , elements of the theory of groups (51), L Berzolari; determinants (30), L Berzolari; linear equations (13), L Berzolari; linear substitutions and linear, bilinear and quadratic forms (28), L Berzolari; rational functions of one or more variables (37), O Nicoletti; general properties of algebraic equations (59), O Niccletti; equations of the second, third and fourth degree and other particular algebraic equations, systems of algebraic equations of elementary type (57), E G Togliatti; methods for the discussion of problems of the second degree and remarks on some of the third and fourth degree (63), R Marcolongo, limits, series, continued fractions and infinite products (45), G Vitali: elements of infinitesimal analysis (101), G Vivanti, relations between the theory of aggregates and elementary mathematics (11), G Vivanti; the analytic functions from an elementary point of view (29), S Pincherle.

The names of the authors of these various articles are a sufficient guarantee of the high standards maintained in the preparation of this very useful work, which was under consideration for more than twenty years but was greatly delayed by the world war and its adverse consequences. From the titles noted in the preceding paragraph it is clear that the term elementary mathematics is used here with a much wider meaning than the one found, for instances, in the

One of the most valuable features of this encyclopaedia is its large number of references to the earliest developments, and hence it is very useful to those who are mainly interested in the history of elementary mathematics. Its object is to give proofs of a number of the most fundamental theorems and simply to give references to the places where proofs of others can be easily found, with an emphasis on the earliest ones. In a few cases one misses here the most recent historical developments relating to the subjects which are discussed. For instance, no references appear therein to the recent discoveries relating to the early partial solutions of quadratic equations by the ancient Babylonians, which were published by O Neugebauer and others in the recently inaugurated periodical entitled

The author index of the present part includes about 800 different names. In view of the emphasis on the earliest developments one would not expect that many of these are those of American mathematicians except when the subjects treated are modern as in the case of the theory of groups. The number of the different names of American mathematicians therein is however more than forty. An international spirit is also exhibited by the fact that the largest numbers of references under individual names appear under those of the two non-Italian mathematicians, A L Cauchy and L Euler. Unfortunately the volume does not contain a subject index, but it is to be hoped that such an index will appear in a later volume of this work. The arrangement of the material is topical under the main headings noted above, and the treatment aims to develop only the more elementary concepts, and hence it is especially adapted to the needs of those who desire to restrict their attention for the time being to the most fundamental ideas involved in a particular subject relating to elementary mathematics.

According to the announced plan the completed work will be composed of three volumes. Volume Il is to be devoted to geometry while volume III is expected to treat a variety of different subjects including the history and the teaching of mathematics. According to the original plan these two subjects were also to be treated in the closing volume of the

On page 43 it is stated that the natural numbers form a group when they are combined according to multiplication while on page 33 it is noted that these numbers do not constitute a group if we assume as a postulate with respect to infinite groups a very useful property of finite groups. It seems to the reviewer very unfortunate that the term group when it relates to an infinite number of elements is not fully defined in the present work, for it would be very desirable to establish uniformity of usage along this line in the modern mathematical literature. On page 31 it is stated that the octic group is generated by three distinct transpositions which is obviously impossible. The number of such oversights is, however, small; the work as a whole can be heartily recommended and it should do much to secure higher standards as regards a knowledge of elementary mathematics. In view of the fact that the mathematical language is so largely international those who know very little Italian will be able to consult this work without much language trouble and by doing so they may be able to improve their knowledge of a language which has become essential to all who desire to keep in touch with the most important modern mathematical developments.

*Encyklopädie der mathematischen Wissenchaften*devotes a little over five thousand pages to the same subject. The treatment in the present volume is however more extensive than that given of the same subject in the*Encyklopädie der Elementar-Mathematik*by Weber and Wellstein.The thirteen main headings of this second part, with the number of pages devoted to each of these subjects and the authors of the articles, are as follows : combinatory calculus (9), L Berzolari , elements of the theory of groups (51), L Berzolari; determinants (30), L Berzolari; linear equations (13), L Berzolari; linear substitutions and linear, bilinear and quadratic forms (28), L Berzolari; rational functions of one or more variables (37), O Nicoletti; general properties of algebraic equations (59), O Niccletti; equations of the second, third and fourth degree and other particular algebraic equations, systems of algebraic equations of elementary type (57), E G Togliatti; methods for the discussion of problems of the second degree and remarks on some of the third and fourth degree (63), R Marcolongo, limits, series, continued fractions and infinite products (45), G Vitali: elements of infinitesimal analysis (101), G Vivanti, relations between the theory of aggregates and elementary mathematics (11), G Vivanti; the analytic functions from an elementary point of view (29), S Pincherle.

The names of the authors of these various articles are a sufficient guarantee of the high standards maintained in the preparation of this very useful work, which was under consideration for more than twenty years but was greatly delayed by the world war and its adverse consequences. From the titles noted in the preceding paragraph it is clear that the term elementary mathematics is used here with a much wider meaning than the one found, for instances, in the

*New International Encyclopedia*, 1923, under the term "Mathematics," where it is stated that "the period of the development of elementary mathematics closes with the seventeenth century." Such a subject as the theory of groups, in particular, was practically unknown in 1700 but its elements are developed in the present encyclopaedia in an unusually clear and comprehensive manner, together with a large number of references to sources.One of the most valuable features of this encyclopaedia is its large number of references to the earliest developments, and hence it is very useful to those who are mainly interested in the history of elementary mathematics. Its object is to give proofs of a number of the most fundamental theorems and simply to give references to the places where proofs of others can be easily found, with an emphasis on the earliest ones. In a few cases one misses here the most recent historical developments relating to the subjects which are discussed. For instance, no references appear therein to the recent discoveries relating to the early partial solutions of quadratic equations by the ancient Babylonians, which were published by O Neugebauer and others in the recently inaugurated periodical entitled

*Quellen und Studien zur Geschichte der Mathematik*. In fact, this periodical is not included in the lists which appear, together with their abbreviations, near the beginning of each of the two parts of the volume under consideration.The author index of the present part includes about 800 different names. In view of the emphasis on the earliest developments one would not expect that many of these are those of American mathematicians except when the subjects treated are modern as in the case of the theory of groups. The number of the different names of American mathematicians therein is however more than forty. An international spirit is also exhibited by the fact that the largest numbers of references under individual names appear under those of the two non-Italian mathematicians, A L Cauchy and L Euler. Unfortunately the volume does not contain a subject index, but it is to be hoped that such an index will appear in a later volume of this work. The arrangement of the material is topical under the main headings noted above, and the treatment aims to develop only the more elementary concepts, and hence it is especially adapted to the needs of those who desire to restrict their attention for the time being to the most fundamental ideas involved in a particular subject relating to elementary mathematics.

According to the announced plan the completed work will be composed of three volumes. Volume Il is to be devoted to geometry while volume III is expected to treat a variety of different subjects including the history and the teaching of mathematics. According to the original plan these two subjects were also to be treated in the closing volume of the

*Encyklopädie der mathematischen Wissenchaften*but up to the present time nothing thereon has appeared as a part of this work. Hence the Italian encyclopaedia under consideration has no modern forerunner along these lines and it will be interesting to see what advances will be made along these difficult and backward lines of mathematical development where permanent progress seems to be most uncertain. In view of the remarkable recent advances made in geometry by Italian mathematicians the volume on geometry will be awaited with high expectations.On page 43 it is stated that the natural numbers form a group when they are combined according to multiplication while on page 33 it is noted that these numbers do not constitute a group if we assume as a postulate with respect to infinite groups a very useful property of finite groups. It seems to the reviewer very unfortunate that the term group when it relates to an infinite number of elements is not fully defined in the present work, for it would be very desirable to establish uniformity of usage along this line in the modern mathematical literature. On page 31 it is stated that the octic group is generated by three distinct transpositions which is obviously impossible. The number of such oversights is, however, small; the work as a whole can be heartily recommended and it should do much to secure higher standards as regards a knowledge of elementary mathematics. In view of the fact that the mathematical language is so largely international those who know very little Italian will be able to consult this work without much language trouble and by doing so they may be able to improve their knowledge of a language which has become essential to all who desire to keep in touch with the most important modern mathematical developments.