I have used index notation to manipulate vectors for almost 40 years. Recently, my principal research collaborator John Gray and I were discussing the method and its origins and why it was not more widely known. This conversation led to a review of past articles and books that used the index method. I have referred to Milne's text many times, it contains much that is useful - but it isn't easy reading. Milne's book (which he began writing in the 1920s) was one of the first books that provided detailed examples of vector and tensor methods in applied mathematics (in the British sense) or theoretical physics - and as a result of changing notation, and the fact that almost every result is demonstrated, it is difficult to read.
The work was inspired by Milne's teacher Sydney Chapman, F.R.S. In the preface, Milne states,
It was he who first expounded to me the view that vectors were not merely a pretty toy, suitable only for elegant proofs of general theorems, but were a powerful weapon of workaday mathematical investigation, both in research and in solving problems of the types set in English examinations. I did not at first believe him; I had been brought up in the idea that, in the words of a distinguished applied mathematician, vectors were like a pocket-rule, which needs to be unfolded before it can be applied and used; similarly I thought (with most others at that time) that vector expressions were a mere shorthand for sets of Cartesian expressions, and that before they could be interpreted they always needed to be translated into Cartesians. Professor Chapman’s reply was; ‘Try for
yourself ’ I had faith in him to do so, and rapidly convinced myself, in spite of my previous beliefs, that he was right. I found further, again under his inspiration, that one could soon learn to think and work that problems, dull or difficult by conventional methods, vectorially gained when treated by vector methods an interest, an ease and a delight which they previously lacked; and that, when a problem was formulated and solved vectorially, the vector solution provided a kinematic picture of the motion in question that gave far more insight into the phenomenon than the corresponding Cartesian solution.
He went on to say,
I began in consequence to use vector methods directly in my own researches, and I began to follow Chapman in lecturing systematically to students through the medium of vectors. The credit for the discovery of the practical advantages of vectors is, however, entirely due to Chapman.
Of course, others would correctly argue that the discovery of the practical advantages of the use of vectors was due to many others including Gibbs, Heaviside, and many others. If you are interested in the history of vector analysis, I recommend reading A History of Vector Analysis by Micahel J. Crowe.
Milne's books reards careful reading. If you want to learn more about vectors and tensors, you will find much that would not be found elsewhere. There is much useful information about the epsilon-delta relationship and a note that about what Milne calls suffix notation. Since it was index notation that took me back to this book, I'll collect some of Milne's comments about suffix (index) notation.
Milne preferred his vector notation to suffix notation (what we call today index notation), though his comments are often at odds with each other. Early on he states
The above notation for the vector calculus is the result of much thought and discussion, and it is hoped that the book may help towards standardizing vector notation. I have at times been criticized as 'clumsy' in my employment of vectors. I will not defend myself against this accusation, but I must defend the notation, which I believe to be concise and convenient. It is incomparably less clumsy and more insight-giving, for the purposes of dynamics, than the suffix notation, which for all its supposed terseness cannot readily describe vector products or lend itself to vector multiplication as an operation. In certain contexts(in Chapter IV), I have deliberately used the suffix notation to secure increased generality, but the results expressed in this notation are not attractive.
However, later he goes on to say
But in more complicated cases the manipulation by use of general suffixes is by far the simplest and safest, and the student is recommended to familiarize himself with this procedure.
Still later, he states
It is clear that the suffix notation is the more convenient analytical expression of the facts of differential and integral calculus contained in the theorem, whilst the vector notation gives the greater physical meaning in each particular case.
It is clear that Milne was ambivalent about index notation - but he provides information about it that is hard to find elsewhere.
So, in summary, this is a difficult book, the fact that notation wasn't standardized when it was written doesn't help. While the book is exhaustive, it is not well organized. I have borrowed this book from the library many times, I've decided that it is time that I owned a copy.
