Levi-Civita: Absolute Differential Calculus

In 1925 Levi-Civita published Lezioni di calcolo differenziale assoluto and, two years later an English translation appeared entitled The Absolute Differential Calculus (Calculus of Tensors). Below we give a version of the Preface to the English edition which was written by Levi-Civita:


(Calculus of Tensors)


Professor of Rational Mechanics in the University of Rome
Fellow of R Accademia Nazionale dei Lincei
Edited by


Authorised Translation by

Miss M Young

Late Scholar of Girton College, Cambridge


The present volume contains a complete translation, made in consequence of a suggestion by my eminent friend, Professor E T Whittaker, F.R.S., of the Italian text of my Lezioni di calcolo differenziale assoluto [Compiled by Dr Enrico Persico (Rome, Stock, 1925)]. Two new chapters have been added, which are intended to exhibit the fundamental principles of Einstein's General Theory of Relativity (including, of course, as a limiting case, the so-called Special or Restricted Theory) as an application of the Absolute Calculus.

I have already had occasion to remark in the Preface to the Italian edition that we possess various systematic and well-written expositions of Relativity by celebrated authors. The short treatment which is offered in the two new chapters of the present work presents some distinctive features which it may be well to point out explicitly.

In the first place, in order not to increase the size of the book unduly, I have thought it expedient to confine myself to tracing the relativistic evolution of Mechanics (properly so called) and of Geometrical Optics, and to developing its most important consequences. In this treatment the whole of Electromagnetism is sacrificed. The sacrifice is certainly regrettable, since Electromagnetism was historically related in the most intimate way to Einstein's conception, having served indeed as the support and model for Restricted Relativity. Furthermore, Electromagnetism, in common with every other physical phenomenon, now comes within the ambit of General Relativity. Much as the omission of Electromagnetism is to be regretted, it has the advantage of reducing the programme to subjects belonging to the pure Newtonian tradition (or to its developments'); and it allows us to take a clearer and more exact view of the transition from the classical scheme of Mechanics to the relativistic one.

For this reason I have followed the method - which I have adopted sometimes in lectures or articles on special subjects of taking the classical laws as the starting point and then of trying to find inductively what modifications - negligible in ordinary circumstances - should be introduced in order to take account of Einstein's ideas; and in the first place, naturally, to take account of his Principle of Relativity, that is to say, the invariant behaviour of these laws under all transformations of space and time, an auxiliary four-dimensional ds2 being duly employed. This method has seemed to me to be preferable to the procedure of enunciating the postulates of relativistic Mechanics in abstract tensorial form, which is so comprehensive in physical content as to be almost inaccessible to ordinary intuition, except with ample comment and illustration.

A further characteristic of our exposition is that we make extensive use not only of geometrical representation but also of the differential properties pertaining to the space-time continuum; attention is drawn also to the special importance of the Einsteinian statics, the treatment being rigorous in some cases, while in others which involve fields variable with the time, it is approximate.

In closing this introduction to Chapters XI and XII I would add that they were prepared, still in collaboration with Professor Persico, at the suggestion of Mr F F P Bisacre, M.A.

In connection with the whole of the English edition, I must warmly thank the translator, Miss Marjorie Long, formerly Scholar of Girton College, who with double competence, scientific and linguistic, has known how to combine scrupulous respect, for the text with its effective adaptation to the spirit of the English language.

I owe hearty thanks also to Dr John Dougall, who, while revising the proofs, has checked the analysis throughout, detected some oversights, and made many useful suggestions for improvement. I wish finally to thank my English publishers, who have not only acceded to, but almost always anticipated, my wishes, in regard to symbols and the typography of the book.


Rome, October, 1926.

Last Updated August 2007