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This is a lovely book: strong emphasis on ideas; a lively sense of humour; a sure logical touch; historical detail that is accurate, relevant, yet quirky (takes some doing!). What’s not to like?

Well, there’s this: it is not easy to decide whether to recommend the book to anybody who doesn’t already know calculus. I’ll return to that. Let me start by describing why this is such a good book.

Firstly, the light touch and the clarity, which together make it wonderfully accessible. Fans of Tom Korner, including yours truly, will be happy to hear that it it as good as his “The Pleasures of Counting” and “Fourier Analysis”, two of the best books on maths ever. Like them, it discusses applications, social context and history but always in a way that supports the maths, which remains the main focus.

Secondly, the balance between rigour and intuition is superbly judged and maintained. Here are two examples: his introduction to differentiation and his handling of the exponential function.

** Differentiation is the first mathematical section of the book, and he starts very, very simply, with 20 019 x 30 016. Ask a mathematician what this is, he says, and “she will instantly reply that ‘to zeroth order’ the answer is 6 000 000 000”. And when asked for greater accuracy “she will be only a little slower to reply that ‘to first order’ the answer is 6 006 020 000”.

That is to say, he introduces differentiation via approximation, to which he gives the delightful name “the art of prediction”, and which he formalises via the little-o notatation $+o(h)$. This is where the balance of rigour and intuition is introduced: he says “I strongly recommend that the reader pronounces it as ‘plus an error term which diminishes faster than linear’.” He leaves it at that, for the moment, with the clear sense that this is a useful way to think and an implied promise that it will be supported by more rigourous argument later on.

** The exponential is introduced only after integration is defined (which itself is presented via a computational approach that is clearly said to be less than general). With integration available, he can define $\log(x)$ for any $x>0$ without worrying about whether $x$ is rational or not. From this he defines the exponential as an inverse: given $y>0$, we determine $x$ such that $\log(x) = y$. In this way, he defines the function $E(y) = x$. He shows that $E$ is its own derivative and sets an exercise to show that it has all the properties of an exponential function. Finally, he defines $\exp(x) = E(x)$ and sets an exercise that goes from showing that $n\log x = \log x^n$ to showing that$\exp(\frac{p}{q}\log(x)) = x^{p/q}$ for all integers $p$ and $q$, provided $q \neq 0$. He uses this to define $x^a = \exp(a\log x)$, explaining that this last step is a brave extension, again with an implied promise that it will be justified later.

Note that the difficulty with defining exponentials for irrational numbers is not avoided but instead is beautifully contained in a context that explains the difficulty. Moreover, the steps from $x^a$ to $a^t$ for some constant $a$, and then to differentiating $a^t$ with respect to $t$ become very simple. He consigns them to an exercise, with a hint that this is an application of the chain rule, so in the end it seems quite direct to find the derivative of $a^t$ by differentiating $exp(t\log a )$.

What you can also see from these two examples is that much of the detail is set in the form of exercises. The whole book has this balance, and it is frankly exhilarating (translation: quite scary!) to see how much he is able to pack into 162 pages of text. Here’s the sequence of topics — remember that this is with emphasis on ideas and with an eye to rigorous proof, of which he reminds the reader frequently by saying he assumes that the functions of which he speaks are “well-behaved”:

• differentiation as prediction, as linear approximation and as tangents
• rules for differentiation
• integration as areas of purely positive functions and then as functions in general with rules for integration and the fundamental theorem of calculus
• growth, maxima and minima, Snell’s law
• logarithms, exponentials and trigonometric functions
• differential equations (as applied to falling bodies, from Galileo on)
• a side chapter I’ll return to, called “Compound interest and horse kicks”
• Taylor’s theorem — with remainder, of course
• the Newton-Raphson method
• 2-dimensional results: maxima and minima, Taylor series
• numerical solutions to ordinary differential equations: Euler’s method, mid-point method
• series (which he calls “infinite sums”)
• the intermediate value theorem, existence of maximum value theorem, mean value inequality

On the one hand, a very recognisable list. On the other hand, the function concept is relegated to simple intuition not in need of great emphasis, actual computation is made central to much of the development, differential equations and numerical analysis are included (indeed central), and many of the details are developed via guided exercises. In these and in many other ways, it is utterly different from the currently fashionable approach imported from the USA.

What I have omitted are some beautiful results that show the importance of analysis as it was developed in the 19th and especially the 20th century: infinite sums that change value when they are rearranged; the topology in two dimensions of maxima, minima and saddle points (charmingly called “islands, lakes and passes”); the existence of transcendental numbers; and much more. The need for rigour is never underplayed, but at the same time the book never flags. If this were a novel, one would say that it moves along at a cracking pace, and many an apparently irrelevant character is later on revealed to be essential to the plot. Take for example the chapter called “Compound interest and horse kicks”. It appears to be a set of examples taken from operations research and only loosely related to calculus and analysis. But it serves the purpose of the book in understated ways: firstly, it shows that calculus can provide (relatively) easy answers that algebra cannot provide at all, and secondly, by demonstrating an interweaving of modelling and analysis, it demonstrates what seems to me the main aim and virtue of the book, which is that rigourously analytical thinking can be of profound value in the most practical of situations.

So, I like it very much, and all teachers of calculus will find material of great value here. Why then am I hesitant to recommend it to readers who don’t (yet!) know calculus? The main reason is that I am really not sure whether such readers should be asked to do all those exercises. Without doing the exercises I think reading the book is of little value. But most beginners studying calculus will not have the technical facility to do the exercises at all quickly, and far from having their eyes opened to beauty and utility of the subject, will become dejected at having to work very hard to proceed very slowly.

In the end, I can only echo Tom Korner’s own advice: read the book for pleasure (yes, it is possible and even desirable to enjoy mathematical exercises!), skimming the bits you don’t appreciate, and come back to it when you’ve done a bit more and find the calculations easier. Indeed, I would suggest going back to it time and again in your mathematical career. It will become more and more of a pleasure to read, and deepen your understanding each time you engage with it.

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