## Henri Fehr's doctoral thesis

Henri Fehr submitted his doctoral thesis

*Application de la méthode vectorielle de Grassmann à la géométrie infinitésimale*to the University of Geneva and he successfully defended it in 1899. This thesis was published as a book in Paris, also in 1899, published by Georges Carré and C Naud. We present below (i) a list of contents in the form of chapter and section headings and (ii) extracts from the review by Edwin Wilson, namely E B Wilson, Review: Application de la méthode vectorielle de Grassmann à la géométrie infinitésimale, by Henri Fehr,*Bull. Amer. Math. Soc.***7**(5) (1901), 231-233.**Application de la méthode vectorielle de Grassmann à la géométrie infinitésimale by Henri Fehr.**

Préface, Introduction:

Rappel de quelques notions de calcul géométrique;

Des courbes gauches:

1° Généralités,

2° Courbure et rayon de courbure,

3° Torsion, formules de Frenet,

4° Courbure normale, formule de Lancret;

De la théorie des surfaces:

1° Généralités,

2° Relations fondamentales;

De la courbure des courbes tracées sur une surface:

1° Théorème de Meusnier,

2° Courbure des sections principales,

3° Formule d' Euler;

De la courbure des surfaces:

1 ° Courbure totale, application aux surfaces réglées,

2° Courbure moyenne; cas particulier,

3° Courbure moyenne quadratique;

Des lignes tracées sur une surface;

1° Systèmes conjugués,

2° Lignes de courbure; théorème de Dupin,

3° Lignes asymptotiques,

4° Lignes géodésiques; courbure géodésique d'une ligne tracée sur

une surface.

**Review by: Edwin Wilson.**

*Bull. Amer. Math. Soc.***7**(5) (1901), 231-233.

The mathematicians Grassmann and Hamilton who almost simultaneously published their first work upon a "calculus of space" were undoubtedly geometers, not analysts - abstract and speculative they may have been - but still geometers. We have only to regret that they were not also clear stylists; for it must be admitted that, with the exception of the attempts of certain persons to show how everything may be done by the methods of vector analysis or quaternions, nothing could be more fatal to the popular acceptation and use of a space analysis than the form in which it was presented to the public by the inventors. Succeeding writers for the most part seem to have erred along the same lines or to have forgotten the stress originally laid upon the interpretation of the analysis. It is therefore with great pleasure that we read M Fehr's little book, which is written with such admirable clearness and selected with such tasteful care that in the compass of ninety-one pages there is included, without the slightest suspicion of crowding, a preface, an introduction on the use of vector analysis, and a fairly complete treatment of differential geometry.

M Fehr originally wrote his book as a thesis to be presented for the degree of doctor of science at the University of Geneva. As a thesis, the work contains nothing original either in vector analysis or in geometry. All the results and methods were known well enough before. Indeed, anyone who has heard such lectures as are given at our leading universities upon these two branches of mathematics ought to be able to put together the material in this book with almost no difficulty. But to put it together in so pleasing a manner is a far harder task.

...

The presentation of the elements of differential geometry given by M Fehr is hardly to be excelled. The usual important and fundamental results are obtained in much less space and time than is customary, and yet with perhaps a gain instead of a loss in clearness. The reason for this unusual brevity is not so much the compactness of the vector notation as the concreteness of the vector idea. No time is wasted in developing analytical formulae. The mind is brought to bear directly upon the geometric questions at hand, and they are solved. It is this constant appeal to visualization that shortens the work; it is this that distinguishes the book from others; it is this that adds so much of perspicuity; and it is this that leads us to recommend the book most heartily to all who teach or study this subject. The vector ideas will clarify and render definite the conceptions of geometry to an extent scarcely possible with other methods. The insignificant amount of vector analysis need cause no fear of difficulty. Mr Fehr is to be thanked for his excellent book ...