Riemann's general metric and a formula of Christoffel constitute the premises of the absolute differential calculus. Its development as a systematic branch of mathematics was a later process, the credit for which is due to Ricci, who during the ten years 1887-1896 elaborated the theory and worked out the elegant and comprehensive notation which enables it to be easily adapted to a wide variety of questions of analysis, geometry, and physics.
Ricci himself, in an article published in Volume XVI of the Bulletin des Sciences Mathématiques (1892), gave a first account of his methods, and applied them to some problems in differential geometry and mathematical physics. Later on other interesting applications, made by himself or his students (to which group I had the privilege of belonging), suggested the desirability of preparing a general account of the whole subject, including methods, results, and a bibliography. This was the origin of the memoir "Méthodes de calcul différentiel absolu et leurs applications" which was compiled by Professor Ricci and myself in collaboration, on the courteous invitation of Klein, and appeared in Volume 54 of Math. Ann. (1901).
There is a chapter on the foundations of the absolute calculus, with special reference to the transformation of the equations of dynamics, in Wright's Tract, Invariants of Quadratic Differential Forms (Cambridge University Press, 1908); apart from this, while special researches based on the use of this method were, continued after 1901 by a limited number of mathematicians, yet general attention was not again directed to it until the great renaissance of natural philosophy, due to Einstein, which found in the absolute differential calculus the necessary instrument vii for formulating the new ideas mathematically and for the subsequent numerical work.
Einstein's discovery of the gravitational equations was announced by him in the famous note "Zur allgemeinen Relativitätstheorie" in the following words: "Sie bedeutet einen wahren Triumph der durch Gauss, Riemann, Christoffel, Ricci ... begründeten Methoden des allgemeinen Differentialkalculus."
In an earlier memoir Einstein had given a new exposition of those elements and formulae of the absolute calculus which more specifically served his purposes. A similar standpoint was subsequently adopted by the most distinguished workers in the field of general relativity, in particular by Weyl, Laue, Eddington, and Birkhoff, all of whom made conspicuous original contributions, both of idea and of method, to the physical theories, in addition to useful and elegant developments of the tensor calculus. Similar statements can be made for Carmichael, Marcolongo, Kopff, Becquerel - to mention, from the vast literature on the subject, only the books I have myself had occasion to consult - while de Donder has avoided the notation of the absolute calculus and used instead the theory of integral invariants.
In recent years there have been some general treatises devoted to the absolute calculus; for instance, those of Juvet, Marais, and Galbrun. Lastly, there is another calculus, in a new order of ideas, not less comprehensive and perhaps even more general, invented by Schouten, and developed with the collaboration of Struik.
In face of this plentiful and valuable literature a new discussion of Ricci's methods might seem to be superfluous; and conceptually this is perhaps true.
In fact, of the improvements and additions to the scheme of 1901 (the memoir in Math. Ann.), derived mainly from the notion of parallelism and on this basis introduced by me into two courses of lectures given at the University of Rome during the sessions 1920-1921 and 1922-1923, all, or almost all, will be found as independent discoveries of the authors already cited, in one or other of their books.
For instance, the definition of a tensor, and some algebraic anticipations of the results intended to simplify the proofs, are to be found in Weyl, Laue, and Marais, all of whom, like Eddington, establish a more or less intimate connection between co-variant differentiation and parallelism. A thorough discussion of the latter is also given by Juvet and Galbrun. But the association with the algebraico-tensorial notation and with the elements of differential geometry is always less detailed and systematic than what I tried to establish in my lectures. The line of argument followed in them has a particular unity, which may perhaps justify their appearance in print at this juncture.
The manuscript was edited with great care and intelligence by Dr Enrico Persico, from notes of the lectures. I wish to express my thanks to him for his valuable help, and to my publisher, Signor Stock (who also attended the lectures), to whose continued encouragement the existence of the book is due.
Rome, December, 1923.
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