Science writer Michael Brooks has thought up a delightful conceit for his latest book.* The narrative takes place in a sixteenth century prison cell in Bologna in the form of a conversation between a twenty-first century quantum physicist (the author) and a Renaissance polymath. What makes this conversation particularly spicy is that the Renaissance polymath is physician, biologist, chemist, mathematician, astronomer, astrologer, philosopher, inventor, writer, auto-biographer, gambler and scoundrel Girolamo Cardano, although Brooks calls him by the English translation of his name Jerome. In case anybody is wondering why I listed autobiographer separately after writer, it is because Jerome was a pioneer in the field writing what is probably the first autobiography by a mathematician/astronomer/etc. in the Early Modern Period.

So what do our unlikely pair talk about? We gets fragments of conversation about Jerome’s current situation; a broken old man rotting away the end of his more than extraordinary life in a prison cell with very little chance of reprieve. This leads to the visitor from the future, relating episodes out of that extraordinary life. The visitor also picks up some of Jerome’s seemingly more strange beliefs and relates them to some of the equally, seemingly strange phenomena of quantum mechanics. But why should anyone link the misadventures of an, albeit brilliant, Renaissance miscreant to quantum mechanics. Because our author sees Jerome the mathematician, and he was a brilliant one, as the great-great-great-great-great-great-great-great-great-great-great-great-great grandfather of quantum mechanics!

As most people know quantum mechanics is largely non-deterministic in the conventional sense and relies heavily on probability theory for its results. Jerome wrote the first mathematical tome on probability theory, a field he entered because of his professional gambling activities. He even included a section about how to cheat at cards. I said he was a scoundrel. The other thing turns up in his *Ars Magna* (printed and published by Johannes Petreius the publisher of Copernicus’ *De revolutionibus* in Nürnberg and often called, by maths historians, the first modern maths book); he was the first person to calculate with so-called imaginary numbers. That’s numbers using ‘i’ the square root of minus one. Jerome didn’t call it ‘i’ or the numbers imaginary, in fact he didn’t like them very much but realised one could use them when determining the roots of cubic equation, so, holding his nose, that is exactly what he did. Like probability theory ‘i’ plays a very major role in quantum mechanics.

What Michael Brooks offers up for his readers is a mixture of history of Renaissance science together with an explanation of many of the weird phenomena and explanations of those phenomena in quantum mechanics. A heady brew but it works; in fact it works wonderfully.

This is not really a history of science book or a modern physics science communications volume but it’s a bit of both served up as science entertainment for the science interested reader, lay or professional. Michael Brooks has a light touch, spiced with some irony and a twinkle in his eyes and he has produced a fine piece of science writing in a pocket-sized book just right for that long train journey, that boring intercontinental flight or for the week in hospital that this reviewer used to read it. If this was a five star reviewing system I would be tempted to give it six.

* Michael Brooks, *The Quantum Astrologer’s Handbook*, Scribe, Melbourne & London, 2017

[Jerome Cardan] was the first person to calculate with so-called imaginary numbers…True.

Jerome didn’t call it ‘i’ or the numbers imaginary,Well, his Latin for them was “sophistica”, often translated “imaginary”.

in fact he didn’t like them very much but realised one could use them when determining the roots of cubic equation, so, holding his nose, that is exactly what he did.Actually, it wasn’t until 27 years after Cardan’s

Ars Magnathat Bombelli connected imaginary numbers to cubic equations. Cardan’s bookdoescontain the first published recipe(s) for solving cubics, andalsothe first appearance of square roots of negative numbers (*), but he never puts the two together.The myth that imaginary numbers first appeared with cubics has great appeal for mathematicians. With the so-called irreducible case of the cubic, we have three real roots but have to make a detour through the complex numbers to find them. As Lipman Bers once said to me, “Well, with quadratics, you just say, ‘the equation has no roots’, that’s all there is to say about it. But with cubics, you have three real roots, yet the formula involves imaginary numbers. So it just makes sense that that was how they were discovered.”

But despite this persuasive imaginary history, the usual (boring, unsophisticated) high-school story happens to be correct.(**) Moral, if you need one: history often doesn’t “make sense”.

(*) Barring some earlier Arab or Indian or Chinese work I haven’t heard of. Actually, Wikipedia suggests that Heron of Alexandria has the first appearance, but I don’t find the argument convincing.

(**) Many historians say that Bombelli’s 1572 work was the first to gave real impetus to the study of imaginary numbers, or something like that. Quite reasonable, IMO.

his Ars Magna…often called, by maths historians, the first modern maths bookActually, I’m a little surprised to see this characterization cited approvingly or at least neutrally, considering the gusto with which you dismantle uses of “The Father of…”.

In many ways the

Ars Magnafits well in the tradition of writers such as Diophantus or even Euclid: geometric algebra, no true algebraic notation. The cubic recipes—it would be misleading to call them formulas—are spread out over many sections, because Cardan is not truly comfortable with negative and zero coefficients, so e.g. x^3+ax=b is treated separated from x^3=ax+b. (This makes his discovery of imaginary numbers even more remarkable.)Before calculus, the invention of modern algebraic notation, chiefly at the hands of Vieta and Descartes, had the biggest impact on the history of European math. If we

haveto nominate a “first modern math book”, I’d go withLa Géométrie. But I’m fine with the usual tag-line, “it’s a process…”I have run across the argument that the solution of cubic equations had an important psychological effect: European math was stepping out from under the enormous shadow cast by Greek mathematics.

The argument to which I’m largely ambivalent, as I don’t really believe in first, although I am prepared to admit that the Ars Magna is historically a very important mathematics book, is that the Ars Magna represents the coming of age of algebra as a mathematical discipline in Europe. Previously it had always been taught as an aid to commercial arithmetic, a status that it also largely had in Arabic mathematics.

I am prepared to admit that the Ars Magna is historically a very important mathematics book,We can certainly agree on that, for a variety of reasons. And Cardan is a fascinating character; I had both his

Ars Magnain translation, and his autobiography (inexpensive edition by Dover books) at one point, but I can’t locate them now.