Vectorial Mechanics, E. A. Milne, Interscience New York, Methuen, London (1948)

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.

Errors of Observation and Their Treatment, J. Topping, Chapman & Hall (1972)

One of the earliest books I remember looking at when I was a freshman at University College Cardiff.  The edition that I have in front of me is the 1972 fourth edition and is a 1985 reprint.  Thus, this is the edition that I originally looked at when learning about error propagation.  A short book at 118 pages, but I prefer it to the now standard An Introduction to Error Analysis by J.R. Taylor.  Topping's book and the similar book by N.C. Barford cover more ground in far less space.  There is much to be said for older books - many of which are around 100 pages in length.  There are newer, longer books, but they don't necessarily improve on the content of the older shorter books.

This book, for me always brings to mind Dr. Ray Hine, who ran the Part 1 labs when I was a freshman in Physics at University College, Cardiff in 1975.  He said, on occasion, that he wanted to make sure that he left the mark of the Physics Department on us as we went forward in life.  In my case, he certainly succeeded, I will never forget his teaching, and his teaching influences mine down to this very day.

Thus this book may not mean as much to you as it does to me, but if you want to learn about error propagation in calculations, this is an excellent place to start.  The fourth edition is still the current edition.  There are many used copies out there, and the Science Paperbacks series was well made.

The Death of Expertise - Tom Nichols, Oxford University Press (2017)

The essential ideas of this book appeared in The Federalist in 2014.  The book is an expansion of the article. The contents of the book are:

Introduction: The Death of Expertise
Chapter 1: Experts and Citizens
Chapter 2: How Conversation Became Exhausting
Chapter 3: Higher Education: The Customer Is Always Right
Chapter 4: Let me Google that for You:  How Unlimited Information is making us dumber
Chapter 5: The "New" New Journalism and Lots of iIt
Chapter 6: When Experts are Wrong
Conclusion: Experts and Democracy

The book discussed the decline of the public scholar, and the rise of the belief that all opinions are equally valid.   A book well worth reading, though if you read the article, you will learn all about the thesis of the book.  The book contains more examples, and explores consequences.  Example include the anti-vaxx  movement, grade inflation, politics, and a host of others.  The section on the Dunning-Kruger effect - the dumber you are the more confident you are that you're not actually dumb - is a highlight.  If you don't read the book, read the article.

Thinking in Systems, Donella H. Meadows, Chelsea Green Publishing. (2008)

Donella H. "Dana" Meadows died in 2001, the draft of this book was completed in 1993.  The draft was edited and published by Diana Wright in 2008.  This book contains a distillation of the work initiated at MIT by Jay Forrester and the MIT System Dynamics group.  Systems thinking is applicable in many areas.  This may well be the best introduction.

The Limits to Growth, Donella H. Meadows, Dennis L. Meadows, Jorgen Randers, William W. Behrens III, Universe Books. (1972)

This book reported on a computer project and reported on the predicament of mankind, it was a report for the Club of Rome.  For many this was the first book of its kind and it was for many, the point where they realized that there were limits to growth.

The idea was not new, I was introduced to the idea by my history teacher Charles B. Fielding.  Mr. Fielding was not famous, nor a well known academic.  As I recall, Mr. Fielding was from Barrow-in-Furness, he was a teacher at several schools in Swansea, Wales.  He was, simply, the best teacher that I have ever known - and he deserves much of the credit for my skills as writer and an analytical thinker.  It is of interest, and says much for the value of a liberal arts education, that the two most influential people in my education were historians.  Charles Fielding was one, the other was the renowned Swansea historian Norman Lewis Thomas, who was at school with my father.  They of course didn't contribute to the development of my mathematical skills, but that is a story for another time.  Mr. Fielding inspired me to read Thomas Malthus; An Essay on the Principles of Population (2nd Edition, 1803) by Malthus is the true source of all work on the limits to growth.

Later work that predates the Club of Rome efforts is that of M. King Hubbert, Hubbert focuses on resources - the Limits to Growth would have been better if it had been informed by some of Hubbert's ideas.

Since this work has been updated many times, I will note that this is a book that everyone who is interested in growth and the limits to growth should read.  It is easy to read because it has been made available on the web.  The original can be legally downloaded from the Donella Meadows Institute.   Since the original work has been greatly modified and updated since it was published, I will save a detailed review for a later version.  At this point, I will say that if you are not familiar with Meadows, you should download this now.  Resource limitations, conservation, and climate change should not be political issues, they are critical to the future that we are creating for our children and grandchildren.

The Art of Modeling Dynamic Systems, Foster Morrison, Dover Reprint of a John Wiley book (1991, 2008)

Long ago, before Amazon.com there was a scientific book club modeled after the venerable Book of the Month Club - which I was surprised to find still exists.  The model was the same, you got four books for a dollar, and then you got a monthly magazine with reviews of other books, and you had to buy four more books at the regular price within a year or two to fulfill your membership agreement.  A selection was scheduled each month, which you got if you didn't remember to send in your postcard saying you didn't want the selection.  There were similar club for long-playing records, 8-tracks, cassettes, CDs and DVDs, most (I actually thought all) are extinct by now.

I'm pretty sure that I got my original copy of this book because I didn't return the reply card, but its possible that I ordered it.  Either way, it is one of the best purchases that I ever made, and I would never have found it without the club, which I think was called the Library of Science, and astoundingly it still exists, I just checked, hence the link.  I guess that many of the old clubs have go online.  Anyway, in 1991 the World-Wide Web didn't exist, I had just got an e-mail account, and things like Gopher were just being developed.  Information was hard to find, and you learned about new books if they were reviewed and when they appeared in reference lists.

Morrison's book, which has the subtitle Forecasting for Chaos, Randomness, and Determinism, was an eye opener.  This is where I learned about modeling and forecasting and that it was possible to study dynamic systems without calculus (I admit it was many years before I realized that there were advantages to this approach.)  The Dover book has a couple of extra diagrams from Morrison's  papers at Federal Forecasters Conferences, and and some added references, but it largely unchanged.  So if you want to learn the basics of modeling and forecasting, this is a good place to start.  The book still doesn't make more than a passing reference the System Dynamics work in Jay Forrester's group at MIT, or the related work of Donella Meadows.  Both groups would have benefited from collaborations, if the system dynamics people had talked to the forecasters, the system dynamics groups would not have neglected to use resources in their models!  I will revisit some of the other work in this area over the next few months.

Modeling the Environment, 2nd Ed., Andrew Ford, Island Press. (2010)

When I'm not working on physics or an administrative task, but am still working, I am most likely to be doing something in the area of forecasting, modeling and data science.  Most of the modeling that I have done has been using Fortran, Python, or Matlab.  I am aware that many others do modeling, environmental scientists, economists, sociologists, and others.  So my question has always been:  How can you do dynamics systems modeling if you have little or no experience with calculus and differential equations.  Well, the answer, in part, can be found in this book.  You can use software that does all the higher level mathematics behind the scenes, and as a modeler, you can focus on stocks, flows, feedback, and timescales.

The book is clear and detailed, and it soon becomes clear that modeling of this kind has its place.  I am going to add the methods to the methods that I regularly use.  As well as describing the modeling process, the book serves as an introduction to two widely used computer tools, Stella and Vensim.  I have been using Vensim as you can download Vensim PLE (Personal Learners Edition) for evaluation or educational purposes at no cost.  The book has more examples in Stella, but translating between the two is relatively straightforward.  I recommend that your familiarize with these modern modeling tools.