‘Brahmagupta’—he rolled the R extravagantly—‘the man who brought astronomy to the Arabs?’ He was seized by enthusiasm now. ‘Binet’s formula? You’re not familiar…I’m sorry, I must not presume. You know of Fibonacci, the Italian? Five, six hundred years ago, he found a sequence that diminishes, or increases, I suppose, depending on how it is expressed—in conformity to a sort of irregular exponentiality. The golden ratio is one to one point six one eight, an irrational number, which is to say it cannot be presented as a fraction. So if you take Fibonacci’s number sequence, any two of those numbers which are consecutive exist in a relationship which is more or less the golden ratio. It makes spirals, Miss Grayling. Isn’t that beautiful? You see, it turns up all the time in nature: Fibonacci demonstrated its effect in breeding rabbits, but I think it might have much wider applications.’

‘Such as?’

‘We already know the sequence and the ratio appear elsewhere: the emergence of branches around a trunk, or leaves around a petiole, but I believe seashells might be another expression. We know they are involute, and that the creature makes the shell that way as it grows. But why are they doing this? In all of nature, everything forms for a reason…

‘And then’—his eyes had closed indulgently but he opened them and glanced at me, fearful he had lost me to boredom or incomprehension—‘consider the notion that the whelks and the turban shells are conical, whereas other shells are not: they merely spiral into themselves without projecting into a diminishing point. Again, Miss Grayling, why?’ He didn’t wait for an answer. ‘Now the best possible illustration of this is the argonaut. I have never had one to use for measurement, but I believe the spiral will conform to the golden ratio, and the relationships between its coils will be Fibonacci numbers. So this is among the chief aims of my expedition.’

I sighed, holding it just short of a yawn. ‘And the map? The spiral on the map?’

‘Ah! Glad you asked. You would understand, I’m sure, that once you start to immerse yourself deeply in a concept, you see its embodiment everywhere you look. Consider the path I have plotted. This’—he stretched forward and pointed at the islands—‘this is the path that your father and the master told me this morning that they want to take: to Wilson’s Promontory and Rodondo Island, then down and into the south-western extent of the Furneaux group, around the south of Clarke Island and on into the Franklin Sound, thus surveying all of the major sealing settlements in one continuous line.’

‘The line makes a spiral,’ I said.

‘It does. Sometimes when I look at it, I see a fish hook which continues in on itself to form a closing spiral. If I were to invert this chart, I believe you would see a bass clef.’