## Reviews of some books by R Marcolongo

**1. Elementi di Calcolo vettoriale con numerose Applicazioni (1909), by C Burali-Forti and R Marcolongo.**

**2. Omografie vettoriali con Applicazioni (1909), by C Burali-Forti and R Marcolongo. **

**Review by: Edwin Bidwell Wilson.**

*Bull. Amer. Math. Soc.*

**16**(1910), 415-436.

In view of the plan that the fourth international congress of mathematicians held at Rome in 1908 should discuss the notations of vector analysis and perhaps lend the weight of its recommendation to some particular system, Burali-Forti and Marcolongo awhile ago set themselves the laudable but somewhat thankless task of collecting and editing all the historical, critical, and scientific material which might be indispensable to a proper settlement of the question by the congress, and this material they published in a series of five notes beginning in the twenty-third volume (1907) of the *Rendiconti *of Palermo and running through several succeeding numbers and volumes. It is needless to observe that the work was accomplished with the expected accuracy. It was, however, not done with all the completeness desirable. The attention of the authors was turned almost exclusively to the minimum system most useful in mathematical physics, that is, to the questions of addition of vectors, of scalar and vector products, of differentiation with respect to a scalar, and of differentiation with respect to space (gradient of a scalar function and divergence and curl of a vector function of position). Considerable discussion was given to quaternions but the fact was fully recognized by the authors that many subjects which might rightfully have been treated were omitted - among which the linear vector function (Hamilton) or quotients and Lückenausdrücke (Grassmann) or dyadics (Gibbs) were perhaps the most noteworthy. The authors not only followed their initial program ; they went further and themselves recommended a particular minimum system of notations essentially like any and all of those now employed but differing in the symbols selected. Thus did it appear that these strivers after unification were prone to follow the path of all unifiers and introduce still greater diversity. Seemingly the universal language of vectors, like the universal commercial language, is destined to suffer constantly new amendments at the hands of its zealots. The conception of unification as conceived in the mind of each enthusiastic unifier appears to be that he shall disagree with everybody and that everybody shall then agree with him. ...

In bringing this review to a close it should be stated that the preponderating length of our adverse criticisms must not be interpreted as a wholesale condemnation of the two volumes. It has doubtless been noticed that the criticisms have been directed against those particular points at which, we feel confident, the authors have made incorrect statements or have unwisely abandoned fruitful algorisms and have thereby left the reader with a wrong or an unfortunately restricted point of view. There is no need to emphasize the excellent features of the work. A large number of these features are common to a considerable number of previous texts on vector analysis ; many of them are new. The fact that there are so many points in which the volumes do not meet our approval is in itself evidence of the value of the books to all students of vectorial methods. In order to acquire a thorough appreciation of a subject it is necessary to examine various methods and points of view, and the restricted or even the wrong ones furnish an amount of instruction which is comparable with that furnished by those that are general and right. ... For the benefit of vector analysis and cognate fields of mathematics we sincerely urge the general study of this work of Burali-Forti and Marcolongo and we especially recommend that each student follow their example and construct the system that pleases him most. That will be the best possible monument to the movement for unification.

**3. Eléments de Calcul vectoriel avec de nombreuses Applications à la Géométrie, à la Mécanique et à la Physique mathématique (1910), by C Burali-Forti and R Marcolongo, translated from Italian by S Lattes.**

**Review by: Edwin Bidwell Wilson.**

*Bull. Amer. Math. Soc.*

**17**(1911), 256-257.

So lengthy a review was recently accorded to two new books on vector analysis by Burali-Forti and Marcolongo that nothing more than the mere mention of the French edition of the first of the two would be needed, were it not for the fact that in the French the authors have added a long and excellent appendix on Grassmann's geometric forms and on Hamilton's quaternions. The object of the appendix is to show the power of the authors' vector analysis by using it to set up the Grassmannian and Hamiltonian systems. There is apparently the further object to set forth these two mathematical disciplines in such a way that mathematicians in general, and in particular those mathematicians who think they know something about the systems, shall be led to conceive or reconceive, as the case may be, these systems as they should be conceived. We have no exceptions to take to the authors' presentation of the subject; it is compelling.

**4. Il Problema dei Tre Corpi da Newton (1686) ai Nostri Giorni (1919), by R Marcolongo.**

**Review by: L Wayland Dowling.**

*Bull. Amer. Math. Soc.*

**27**(1921), 256-257.

This little book, in the well known style of the Hoepli manuals, presents, as its title indicates, an account of the problem of three bodies from the time of Newton to the present. The author has limited himself strictly to a descriptive account of what has been accomplished in this interval of time, with full references to original memoirs and papers where the interested reader can find the complete developments. Professor Marcolongo is well known as an authority in the field of dynamical systems, and this book from his pen will be welcomed by all who are interested in the development of mathematical astronomy. Here will be found references to the works of over 200 authors who have contributed to one or more phases of this celebrated problem, together with a short description of the aim, the method of attack, and the results attained. ... The complex development of modern mathematics calls for more books of this type: mathematical Baedeckers, without symbolism, with concise statements of aim, method of attack, and results, and with full references to original sources.

**5. Leonardo Da Vinci, artista-scienziato (1939), by Roberto Marcolongo.**

**Review by: Anacleta Candida Vezzetti.**

*Books Abroad*

**18**(1) (1944), 89.

The literature on Leonardo is immense. Artists and men of letters, critics and historians, have written about this excellent artist, this magnificent man, learned writers have evaluated his importance as a scientist. But his entire encyclopaedic creation has rarely been presented in one volume. This book, which was published in the year when the great Italian was honoured in Milan with a superb "one-man" exposition, presents him in his manifold artistic and scientific activities, as a phenomenon of the proud period of the Italian Renaissance. The first four chapters tell his life, and give the history of his paintings, sculptures and scientific discoveries. In what follows, the characteristics of his art and the extraordinary adventures of his manuscripts are considered. The final chapters furnish a synthesis of his scientific investigations. The presentation is easy and agreeable, free from intricate formulae, technicalities and abstractions. Any person of culture can understand and enjoy the book. It is a tribute to the Hoepli Publishing House that issued a work of such timeless value under the Mussolini regime.

JOC/EFR May 2018

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