The opening chapter is set in 1828, when Gauss unwillingly sets off on his first trip in years from his home town of Göttingen to attend a conference in Berlin. This trip concludes in a comedy of failure when Humboldt attempts to have an (anachronistically early) Daguerreotype taken of their encounter. Measuring the World then proceeds in roughly chronological chapters that alternate between the two protagonists, describing their educations and lives and work.
Von Humboldt and Gauss are presented as a study in contrasts. Both were prodigies, but one came from an aristocratic family and was given a rationally planned education along with his brother, the other came from a poor working-class background. One wandered the world while the other stayed at home and investigated the universe from there. (A major theme is the nature of space and time and their measurement.) And so forth. Only towards the end of the novel, as a correspondence between the two starts to merge with their streams of consciousness, is there a convergence in their approaches to ageing and the approach of death.
Kehlmann takes some liberties with his history — I don't think there is any evidence that von Humboldt's elder brother attempted to kill him, for example — and there is some outright magical realism, with Gauss given an ability to see into the future. The bigger distortion is that the personalities of the protagonists are caricatured for comic effect: the contrast between Humboldt's asexuality and Gauss' lecherousness is continually played up, for example. I'm not that taken by this in a historical novel: a good fraction of the people reading a novel like this will know little or nothing about Gauss or von Humboldt and are likely to go away with a rather distorted, if not actually wrong-headed, view of them.
Kehlmann does something to mark this by exploring the construction of reputation: we see von Humboldt deliberately moulding his own in the editing of his journals, while Gauss refrains from publishing ideas he thinks the world isn't ready for. And there are other subtle ways in which the narrative undercuts its own claims to accuracy. Though Measuring the World is largely dialogue-based, it has no direct speech at all: it is entirely in indirect speech, which gives a shimmer of uncertainty to everything. This works much better in German, which allows this to be marked off from the narrative with a minimum of effort; the English translation seems quite decent to me, but in following this stylistic feature is unavoidably somewhat stilted.
These qualms aside, Measuring the World is a lot of fun and I really enjoyed it: as a learner reading it in German I was grateful for the motivation provided by the comedy. (The bibliographic details below are for an English translation, but I read this as Die Vermessung der Welt, Rowohlt Taschenbuch Verlag 2013.)