30 December 2020

Victorian Arithmetic

Last term I received an email from Peter Elliott, the great grandson and biographer of Sir Thomas Muir (1844 to 1935). Thomas Muir was a renowned Victorian mathematician, principally associated with the Theory of Determinants, on which he wrote copious volumes. He was also Superintendent General of Education of the Cape Colony in the early 1900s. Muir wrote A Text-Book of Arithmetic for Use in Higher Class Schools in 1878, when he was the mathematical master in The High School of Glasgow. An online copy of the textbook can be accessed here on Google Books. 

Readers of my blog will know that exploring old maths textbooks is a hobby of mine. I have a page full of links to old maths textbooks here, and I own a large collection of physical books dating from the mid-1800s to the late 1900s. I have blogged about old textbooks numerous times, including in the following posts:

Extracts from old textbooks also feature heavily in my book, A Compendium of Mathematical Methods.

I promised Peter that I would have a look at Muir's textbook and share my findings with my readers. The book is rich in content and gives us a wonderful insight into how maths was taught 140 years ago. I strongly recommend that my readers take a look through the textbook themselves. In this post I have picked a few highlights, to give you a taste of the kind of thing you can expect to find in a textbook from this era.

As is typical of textbooks from the 1800s, the explanations are set out in prose. Modern mathematics textbooks tend to explain concepts and procedures by way of annotated examples, as opposed to heavy paragraphs, so the wordy nature of old textbooks comes as a surprise to us. The exercises are also rather different in style to what we are familiar with. Interspersed throughout each chapter, the challenge level in these exercises is high. To give an example, in the first chapter ('the nomenclature and notation of integral numbers'), we are presented with the following tasks:
Everything is taken to a far greater extreme than it is now. Check out the length of the numbers that students are asked to say out loud. They seem almost comical to us now.

Later in the book, we have a similarly amusing set of addition questions. In the days before calculators existed there were certainly some jobs - in the counting house for example - which required such lengthy manual additions. Just imagine the monotony of performing such arithmetic. 

One thing that makes Muir's textbook slightly different to comparable textbooks from the same era is his inclusion of 'real-life' example, as we can see here:
Muir also presents us with sets of questions that, apart from the seemingly cumbersome numbers chosen, would not be out of place on variationtheory.com.

The vocabulary in this textbook is similar to other textbooks from this era. I have previously presented at conferences on the evolution of maths vocabulary. One example is the interchangeable use of the words cipher, nought and zero. Another example is the language associated with powers:
These days it is rare to hear this language (i.e '4 power 3' for the third power of four. We are more likey to say 'four to the power of three'.). Also, note that this book pre-dates the time in which textbooks started to incorrectly and confusingly claim that the index can also be referred to as a power. 

Other vocabulary that has fallen out of common usage in schools include 'measure':

There's a bit more on the word measure here, and the alternative 'sub-multiple' is offered:

For prime factor decomposition, modern textbooks almost always feature prime factor trees. Old textbooks tend to feature repeated division. Interestingly, this book has a slightly different approach:
The author acknowledges that for large numbers, the process of finding the prime factors can be laborious. The example given is 9211, which resolves into 61 x 151.

I admit that I hadn't realised there was a difference between a denomination and a denominator. As far as I know, the term denomination has fallen out of use in the context of fractions.

I like it that mixed numbers are properly defined in this chapter. The attention to detail in definitions and notation is something that has been lost over the years. These days, too much is assumed.

Interestingly, the author refers to improper fractions as 'sham' fractions with an 'integer in disguise', and tells us that we should express them in their proper form (i.e. as mixed numbers).

The fractions chosen for the fraction addition exercise fascinate me. I like question five - the powers in the denominators seem to be a clever task design strategy, drawing attention to sensible denominator choice.

In all my journeys with old textbooks, something I haven't seen before is the idea of approximate simplifications of fractions.

This is interesting. At first glance I thought this was totally pointless. But I get the idea - if we want an idea of the magnitude of a fraction, we don't need to be exact.
Here's an exercise on this:

The author goes on to say, "There is another mode of finding approximate simpler forms of fractions which is in itself more important and more interesting than the above.' He then delves into continued fractions, which represents a massive step up in complexity compared to the rest of the book so far.

Victorian Contexts
There is a large section of the book dedicated to mixed 'practical examples'. It is often these contextual examples that provide the most amusement and intrigue: they give us a glimpse of what Victorian life was like. Here are a few examples, featuring disease, quills, boys doing arithmetic, and wine.

I hope you enjoyed this little tour of a Victorian maths textbook. 

If you wish to learn more about the life of Sir Thomas Muir, do check out Peter Elliott's recently published book Thomas Muir: ‘Lad O’ Pairts’: The Life and Work of Sir Thomas Muir (1844–1934), Mathematician and Cape Colonial Educationist


  1. Fascinating! Thanks for sharing this. I really like the use of continuous prose in teaching some of these ideas. The language part of CPAL is an important but often overlooked modality. I started using more written language organically in the early days of tutoring online and stuck to using it ever since. I had to communicate a lot more with voice and through type text before online whiteboards were developed. So it is reassuring to see this use of language in these old books.

    Fascinating also to see multiple bases in use. After CPD and reading Paul Lockhart's book Arithmetic, I've been really intrigued with multi-base and cultural number systems. I've been teaching multi-base to home schooled tutees using exploding dots etc. Thinking across bases seems to build great flexibility and generalising skills. Pre-1971 children would have had to learn to calculate in mixed-base with shillings, old pence and pounds anyway. I ought to explore these old arithmetic textbooks as well.

    1. Thanks for your comment Atul. Agree about multiple bases - not something I came across myself until I became a teacher, but use to be widely taught.