## Reviews some of Gomes Teixeira's books

The three books by Gomes Teixeira, of which we give extracts of reviews below, are written in Portuguese, Spanish and French, respectively. We quote from one review of each work which appeared in the

*Bulletin of the American Mathematical Society*.**1. Corso de Analyse Infinitesimal (3 vols) (1892-96), by F Gomes Teixeira.**

**1.1. Review by: James Pierpont.**

*Bull. Amer. Math. Soc.*

**5**(10) (1899), 483-484.

While perusing the present book it was a constant source of regret to me that Portuguese is not better known in our country. Otherwise this admirable work on the calculus would enjoy widespread popularity among us. Its author, the distinguished director of the Academia Polytechnica at Porto, has been uniformly successful in the difficult task of selecting from the immense material available. The manner of presentation leaves nothing to be desired. The style is lucid and elegant, and the whole work bears in a refreshing manner the imprint of an original mind. In many places the author has incorporated parts of his own prolific and valuable writing on the subject. In regard to rigor, it seems to us that Professor Teixeira has very happily chosen the golden mean. The excessive rigour of a Weierstrassian has been wisely avoided; at the same time the author has given this matter due attention. An occasional slip will doubtless be corrected in later editions. Altogether the work has so favourably impressed us that we should prefer to see it translated into English rather than any other work on the subject we know of. It is a deplorable confession that the English language does not today possess a work on the calculus of this class.

**2. Obras sobre Mathematica (Vols. I and IV) (1904-08), by F Gomes Teixeira.**

**2.1. Review by: Anon.**

*The Mathematical Gazette*

**4**(76) (1908), 398-399.

The major part of these handsomely printed quartos [Volumes I and IV] is concerned with the developments of functions of various kinds in series. For instance, the first hundred pages of Volume I comprise a series of papers on Taylor's Theorem in the case of functions of both real and complex variables, first in an elementary treatment and then by the methods of Cauchy, Riemann, Weierstrass and Mittag-Leffler. The paper is completed by a discussion of the series of Burmann and Lagrange with a generalisation of the former. The memoir on curves having the same evolute as the ellipse, was crowned in 1898 by the University of Belgium. Memoirs on the convergence of certain formulae of interpolation, and articles on infinitesimal analysis form an important part of the volume, which also contains matter of a more elementary kind in connection with plane analytical geometry. Volume IV is a revised and enlarged edition of a work entitled 'Tratado de las curvas especiales notables', crowned by the University of Madrid in 1899. It apparently owed its existence to a suggestion thrown out by M Haton de la Goupilliere in the early days of the 'Intermediaire'. It is now translated into French. Another volume will be necessary to complete this great undertaking, and any reader who has the whole of Dr Texeira's work in his hands, as well as the classic treatise of Gino Loria, will have at his disposal the results of an enormous amount of research. ... The next volume will deal with equations which according to the parameter involved represent either algebraical or transcendental curves - parabolas and hyperbolas of any order, cycloidal curves, and the like. Two chapters will be devoted to gauche curves, and one to the theory of Poinsot's polhode and herpolhode. The book will be a perfect storehouse of material for examination questions on the applications of the calculus to Analytical geometry.

**3. Tratado de las Curvas Especiales Notables (1905), by F Gomes Teixeira.**

**3.1. Review by: Charles H Sisam.**

*Bull. Amer. Math. Soc.*

**13**(5) (1907), 249-250.

This volume had its inception in a prize problem proposed in 1892, and repeated in 1895, by the Royal Academy of Sciences of Madrid, requiring "An orderly list of all the curves of every kind to which definite names have been assigned, accompanying each with a succinct exposition of its form, equations and general properties, and with a statement of the books in which, or the authors by whom, it was first made known." This programme our author has closely adhered to except in one particular. To attempt to give the properties of all such curves would be extremely difficult and would make the resulting work unwieldy, he has therefore wisely limited himself to a list of over one hundred curves so selected as to include almost all of especial importance. ... The treatment is, throughout, quite elementary, and can be followed by anyone with a knowledge of analytics and calculus. The book is not, however, suitable for a student seeking a systematic treatment of the theory of the higher curves, as its aim is the consideration of noteworthy curves and not the systematic exposition of curves in general. It is a treatise, not on curve theory, but on particular curves. The method of exposition, whenever practicable, is as follows : the rectangular and polar equations of the curve are given, the form of the curve is derived from the equations and a figure of the curve is shown. The most interesting geometric properties of the curves are then deduced, the parametric equations - when the curve is unicursal - derived and the integrals for the length of arc and the area of the curve obtained.

**4. Obras sobre Mathematica (Vol. V) (1904-08), by F Gomes Teixeira.**

**4.1. Review by: Anon.**

*The Mathematical Gazette*

**5**(88) (1910), 342-343.

This volume is a translation into French of the second part of the author's "Treatise on Plane and Gauche Curves," the first part of which has already been noticed in the

*Gazette*. It forms a fitting complement to the rich store with which we were presented in its predecessor. Dr Teixeira has laid under tribute practically the whole of available mathematical literature. Vast as is his display of erudition, he has marshalled, modernised, and presented his material with great skill. And when we note, as is often the case, the elegance of his methods in continuing or completing an investigation, we could lament that the time the author has spent upon analysis has not been devoted to geometry. ... Dr Teixeira is to be congratulated on the manner in which his Government and the University of Madrid have recognised the value of his labours, and we, too, must congratulate him on the successful termination of his task.

**5. Sur les Problèmes célèbres de la Géométrie élémentaire non résolubles avec la Règle et le Compas (1915), by F Gomes Teixeira.**

**5.1. Review by: Raymond C Archibald.**

*Bull. Amer. Math. Soc.*

**24**(4) (1918), 207-210.

There have been many historical surveys of the three famous problems of the ancients. One such was Montucla's anonymous work of 1754 on the history of the problem of the squaring of the circle with a supplement concerning the problems of the duplication of the cube and the trisection of an angle. But a more adequate history of the problem of the duplication of the cube was published by Reimer in 1798. An accurate and still more elaborate presentation which took due account of later research was published about a century later by Ambros Sturm. ... Since Montucla's work is very scarce, those unacquainted with German or Italian who wished to learn the main facts in such surveys as the ones to which I have referred, have had, till recently, considerable difficulty in satisfying their desires. We have now, however, the very interesting and excellent volume under review, of Professor Gomes Teixeira, Rector of the University of Porto. His power of lucid exposition and his scholarly style are probably familiar to many Americans through the two-volume 'Traité des Courbes spéciales remarquables planes et gauches' of 1908-09. Nearly the whole of the volume on "problèmes célèbres" is given over to a consideration of the three famous problems of the ancients. Chapter I is entitled: "Sur le problème des moyennes proportionnelles. Duplication du cube;" Chapter II: "Sur la division de l'angle;" Chapter III: "Sur la quadrature du cercle;" and the last chapter: "Sur l'impossibilité de la résolution par la règle et le compas des problèmes considérés précédemment." There are many references to the author's treatise on curves and it is especially in this connection that new features are introduced. ... We heartily recommend Professor Gomes Teixeira's book for every mathematical library, as no other publication of the kind can take its place. The little book is characterized by marked individuality.

**5.2 Review by: Anon.**

*The Mathematical Gazette*

**9**(128) (1917), 57.

This sumptuous monograph discusses each problem in turn in general terms, followed up in each case with the more famous "solutions." For instance, we find seventeen sections giving typical attacks on the duplication of the cube, dating from Plato and Archytas to Clairaut (1726) and Montucci (1869). The last chapter deals with the impossibility of the solution of the famous problems by rule and compass alone. The contents of this chapter are well up to date, and include a neat proof of the general irreducibility the equation x

^{n}= b, due to the author and published in the Spanish

*Revista*in 1914. There are one or two obvious misprints, but they do not interfere with the easy flow of Professor Teixeira's exposition. From the title it gathered that the monograph is written in French.

**5.3. Review by: P E B Jourdain.**

*Science Progress*(1916-1919)

**12**(45) (1917), 160.

This exceedingly useful compilation is a very complete summary of the various solutions that have been given of the three celebrated problems: the duplication of the cube, the trisection or multisection of the angle, and the quadrature of the circle. The contributions of each mathematician are dealt with in separate sections and in an analytical manner, so that modern methods and notations are used throughout. Such a method certainly conveys a clear idea of the point at issue to modern students, and perhaps this indicates the object of the book. The problem and solutions of the duplication of the cube are traced from Hippocrates to Montucci (1869), those of the division of the angle are traced from Hippias to Kempe, and those of the quadrature of the circle from the approximations given in the Rhind papyrus to the expressions of Mansion (1910). A fourth chapter is on the impossibility of the resolution of the above problems by the ruler and compass, and is a very useful analytical treatment from a modern point of view.