Important books: Mathematics for the million by Lancelot Hogben
As a new regular feature of the HPM Newsletter, members of the HPM community will be asked to name a book (or books, or a paper) that has been important to them and to give their reasons for this. In this way, “classics” may be introduced to new audiences. In this issue, we ask recently retired HPM Newsletter editor Chris Weeks for his choice of book.
Lancelot Hogben, Mathematics for the Million, London, Allen & Unwin, 1936.
I was asked recently how I had become interested in the history of mathematics. I was always vaguely aware that mathematics and all science must have been developed over time but I owe a more specific awakening of interest in the history of mathematics to Hogben’s Mathematics for the Million. I read it while starting the more advanced mathematics courses at school and for the first time I found a book about mathematics that was not a recipe book for manipulating mathematical expressions. (To be fair a little more than following rules is required for higher school mathematics, but not much more.) The striking thing about the book is that the mathematics is presented from the outset as an historical development. He says:
The customary way of writing a book about mathematics is to show how each step follows logically from the one before without telling you what use there will be in taking it. This book is written to show how each step follows historically from the step before and what use it will be … [author’s italics]
Hogben claims to assume in the reader no more than an elementary knowledge of mathematics and he certainly starts from number and simple geometry. But a familiarity with algebraic manipulation is necessary almost from the outset. By the end of the book we meet almost all the mathematics that was then part of advanced mathematics in British schools prior to university.
The book was published by Allen & Unwin, London in 1936 and when I read it about 20 years later copies were to be found in almost every public lending library. It was endorsed by H G Wells (‘a great book of first class importance’) and by Albert Einstein (‘makes alive the contents and elements of mathematics’). There was a new edition in 1968 and after going out of print it was surprisingly reissued in paperback format by the left wing publisher Merlin Press in 1995 but is no longer in their list. How does it stand the test of time?
The first thing I noticed on rereading it was the confident and optimistic tone of the writing. It is unashamedly Whiggish in style. Science has made enormous progress since the Stone Age and mathematics is the tool that allowed such progress. Hogben also claims that mathematics is essentially a language about size. It enables us to calculate and measure. He is dismissive of the Plato school of philosophical reflections about the nature of number. That line only leads to mysticism. Mathematics is no use unless it makes things happen. As far as Greek science is concerned, the centre is Alexandria and Hogben is careful enough to insist on the description Alexandrian and to point out that the only thing that the mix of peoples associated with Alexandria had in common was their use of the Greek language.
The book claims to teach mathematics through tracing its story of development. In fact it sets out to be a teaching book (the author advises us to have paper and pencil to hand). It is carefully constructed in terms of increasing levels of mathematical demands and starts with simple arithmetic and geometry and ends with applications of the calculus and a final chapter on statistics and probability. In keeping with its time, there is a quite a lot about the use of determinants but no appearance of a matrix.
For my part, when I used the book, it almost exactly mirrored what I was studying at school but with the advantage of a wider context, more interesting problems and some indication of history. In fact the economic or military stimuli to new mathematical discoveries is evident. But there was richer fare including a whole chapter on ‘Mathematics for the mariner’ which includes a description of the celestial sphere and an introduction to spherical geometry.
For its mathematical content, I would certainly recommend the book to any budding mathematician. Of course the reader has to learn to read mathematics presented in a slightly unfamiliar way, but that’s no bad thing. And the reward is a number of delights, including this instruction from the British Ministry of Agriculture and Fisheries (1935) on how to lay out a right angle for a plantation of fruit trees:
The 24th link is pegged at the point from which the right angle is to be set out, the nought end of the chain and the 96th link are pegged together, back along the base line, so that the piece of chain 0-24 is taut. If the 56th link is taken in the direction required until both the sections 24-56 and 56-96 are taut, then the point reached will be at right angles to the base line.
So we have a 24:32:40 or 3:4:5 triangle. The reader of Hogben’s day would know that standard land measurement used Gunter’s chain of 100 links, where 1 chain is 22 yards (the length of a cricket pitch, as every schoolboy knew) and where, of course, 10 chains = 1 furlong and 8 furlongs = 1 mile. Today’s reader may need to do a bit of research but Google is at hand and a calculator will make the helpful addition of logarithmic and trigonometrical tables in the book unnecessary (although it might be instructive to learn how to use them).
Finally, a word about Hogben himself, which would have impressed me even more had I known it at the time I read his book. Hogben, largely self-educated at Stoke Newington Public Library, was an experimental zoologist and medical statistician by trade. During the First World War he served as an ambulance driver for the Red Cross in France but after his decision to leave the Front and return to Cambridge he was imprisoned in 1916 as a conscientious objector. He married the mathematician and feminist Enid Charles, a statistician who worked on fertility rates. Both spoke against the then fashionable idea of eugenics. Hogben also wrote Science for the Citizen (1938), a companion book to Mathematics for the Million in the collection ‘Primers for the Age of Plenty’, a project which he edited and which was intended to encourage in his readers the self-education that he had valued in his youth. But both books, enormously popular as they were, were incidental to his main work in biology and statistics. The Hogben archive is housed at Birmingham University.
[For biographical information I have made free use of the Wikipedia entry for Lancelot Hogben]