## Reviews of Ernesto Pascal's texts

**E O Lovett**, Review: Repertorio di matematiche superiori. I (Analisi), by Ernesto Pascal,*Bull. Amer. Math. Soc.***5**(1899) 357-362.In no subject is special specialization growing more imperative than in mathematics; in the midst of difficulty and demand the student should hail with delight the valuable services of a work so admirably adapted to purposes of orientation as Professor Pascal's repertorium promises to be. The author's plan, adhered to without deviation, is to present with regard to each theory of modern mathematics the fundamental definitions and notions, the characteristic necessary theorems and formulae, and citations to the principal works of its bibliography. The definitions are clear and unequivocal; the statements of the theorems, always given without demonstration, are concise and unambiguous; and the bibliographical references are sufficiently full to supply the needs of the general mathematical reader.

**J K Whittemore**, Review: Die Variationsrechnung, by E Pascal,*Bull. Amer. Math. Soc.***6**(1900) 352-354.The German translation of the "Calcolo delle variazioni" published by Ernesto Pascal in 1897 gives to American mathematicians in convenient form the best book on the calculus of variations that has, to our knowledge, appeared up to the present time. The book consists of only 150 octavo pages and presents concisely the principal facts of the subject. A valuable feature of the work will certainly be found to be the very excellent and apparently complete bibliography given in connection with brief accounts of the development of the calculus of variations. No book with which we are acquainted could be better adapted to controvert the lay opinion that everything in mathematics is exact and beyond dispute. Again and again the author calls attention to errors made by writers in this field. He often calls attention, too, to gaps which remain still to be filled in the theory, and makes the reader sometimes feel that the results which we already have rest on a rather precarious foundation. The chief fault of the book, from our point of view, is that it sacrifices simple and natural discussion to the pursuit of the end so dear to Italian mathematicians, the greatest possible generality. The apparent purpose of the author is to give an account, absolutely rigorous as far as it goes, of the present condition of the science. That such an end is in the calculus of variations especially difficult to attain appears from the fact that the proofs are not always precise and that the author prefers often to tell us that the work given is not rigorous rather than to attempt to make it so.

**Anon**, Review: Repertorium der Höheren Mathematik (Definitionen, Formeln, Theoreme, Literatur). Part II Geometrie, by E Pascal,*The Mathematical Gazette***2**(35) (1902), 219.The advanced student and the teacher will welcome this translation into German (by Ober Lieutenant Schepp) of the second volume of the Synopsis of higher mathematics published two years ago by Professor Pascal of Pavia. Here he will find, without the labour of turning up a number of journals or text-books, a definition, a formula, a theorem, and a perfect treasure-house of bibliographical information.

**E B Elliott**, Review: I Gruppi Continui di Trasformazioni (4th edition), by Ernesto Pascal,*The Mathematical Gazette***2**(38) (1903), 264-267.The little book on Lie's theory which is before us deserves a hearty welcome. For a short time longer there is still no English book on the subject. Let those of us who know a little Italian peruse the present manual. It is all the easier to start upon because there is not room in it for the dignified style and the almost wearisome elaboration of the greater works brought out under Lie's own auspices. Few authors know so well as Sig. Pascal how to present higher mathematics in didactic form. The range of his mathematical learning is moreover cyclopedic. The rate at which one useful and up-to-date Manuale Hoepli from his pen follows another is remarkable. Signs of haste in production, though not entirely absent, are rare. The Manuali Hoepli are books of size for the pocket. Two pages would go on one of an ordinary octavo. The type, which is beautifully clear, is almost extravagantly large. The purchaser for half-a-crown of the present volume might well be forgiven for expecting only a meagre sketch of first principles, and not much of the analysis, perforce abounding in triple suffixes, etc., which is formidable of aspect even on the ample page of Teubner's Lie-Engel. He will be agreeably disappointed. The work is not unambitious. Its aim is "Without lack of rigour and generality to contain in little space all that forms the basis of this advanced part of pure mathematics." The author has at any rate succeeded in making clear in their complete forms the principles and processes of the general theory. Special theories, and in particular the whole subject of contact transformations, are reserved for a promised further volume.

**A Dresden**, Review: Lezioni di Calcolo Infinitésimale Part 1 (4th edition), Part 2 (4th edition), by Ernesto Pascal,*Bull. Amer. Math. Soc.***28**(1922) 315-317.In his preface, written May 1917, Pascal says " It is certain that through the profound changes which the critical spirit has made in the foundations of the calculus, even a course intended for those for whom mathematics is a means rather than an aim, cannot but use the new results which have been reached . . . it would therefore exhibit a shortsighted view and little esteem for the ability of the future engineer, to believe that it would be sufficient for them, at least if they can, to learn to operate the calculus in about the way in which a workman knows how to operate a machine made by others, and of which he does not know the inner connections."

**C H Sisam**, Review: Repertorium der Höheren Mathematik, by E Pascal, Vol. II (Geometry), Part 2 (Geometry of Space) (2nd edition),*Bull. Amer. Math. Soc.***29**(1923) 373.The first section of the volume on analysis and also of the one on geometry of this second edition of Pascal's Repertorium of Higher Mathematics appeared in 1910. They were reviewed in this BULLETIN (volume 19, pp. 372-374). The second section of the volume on geometry, after being on the eve of publication for nearly ten years, has now appeared. This section is devoted to geometry of space. In a subject so extensive, the first problem is a proper selection of topics for consideration. Somewhat more than half of the work is devoted to algebraic surfaces. Of the remainder, about one third is devoted to algebraic curves, a third to differential geometry and the remainder to line geometry and algebraic transformations of space. ... The treatment of the various topics is always good and occasionally excellent. ...

**L S Hill**, Review: Repertorium der Höheren Mathematik, Vol. I (Analysis), Part 2 (2nd edition), by E Pascal,*Bull. Amer. Math. Soc.***35**(1929) 737-738.In conformity with the general plan of the undertaking, the present section, like its predecessors in the second edition, attains a definitely higher degree of readability than one has hitherto been accustomed to expect in professedly encyclopedic summaries of mathematical science. Although proofs are, of necessity, frequently omitted, theorems are stated with clearness and accuracy, and in well-arranged sequence. References to the literature, while by no means exhaustive, will in most cases provide ample orientation for those who wish to undertake specialized studies. By reason of the arrangement of the material and the style of exposition, it is probable that most mathematicians will find in this section of the Repertorium not only appreciable stimulation but also a measure of downright enjoyment.

**T H Gronwall**, Review: Repertorium der höheren Mathematik Vol. I (Analysis), Part 3, by E Pascal,*Bull. Amer. Math. Soc.***36**(1930) 31.This is the final part of the Analysis volume of the second edition of the Repertorium] the first part appeared in 1910 and the second in 1927. ... All of the authors have very successfully achieved their purpose of presenting concisely the main results of their subjects together with references to the more important literary sources, and the book will undoubtedly prove indispensable to the workers in any of the fields treated.

JOC/EFR May 2013

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