Substance: This "sequel" to the seminal satire "Flatland" falls flat. The science as explained here is rudimentary. Burger explains in detail the things we already know about 3D space, and gives no new insights into higher dimensions other than to propose they really do exist - somehow. The visualization experiments work only when the author goes back to Lineland for his analogies. The story-line is boring and repetitious. The milieu is confusing and improbable even in the context.
Style: Boring. In Burger's defense, he probably has not read the many excellent SF novels predicated on the perceptions of people confined to artificial environments or alien landscapes.
NOTES: (despite the overall negative impression, there were a few interesting comments)
p.54-55: the experiences of the travelers who, on a bet, head east and west and meet each other again was the most interesting story in the book, even though it reeked of Colonialist rhetoric.
p. 98-99: gratuitous dig at humanity and its "criminality".
p. 134 ff: the extended description of the trials and tribulations of a scientist whose observations differ from the consensus opinion of the establishment - and who is eventually vindicated when others replicate his measurements - is a good analysis of what goes wrong when political concerns trump scientific investigation. In particular,
p. 134: i when results differ from expectations, one should investigate what, if anything,was different in the methodology this time.
p. 138 (Burger attributes all the outstanding insights to a precocious child): "I had to think of something peculiar to answer a peculiar question".
p. 143: an opponent argues that "If a series of calculations is in conflict with the first fundamentals of mathematics, the series is false." By extension, any result that contradicts what is KNOWN TO BE TRUE (a priori, or by logical reasoning) MUST BE FALSE (hello, Climate-gate!)
p. 160: the protagonist keeps secret his meetings with the Sphere for fear of being thought to communicate with "evil creatures from the spirit world" - but how is that different from communicating with unknown creatures from an invisible world?
p.169-170: The Sphere, inexplicably, divulges his information in minute snippets at annual intervals (for which there exists no really meaningful plot rationale); also, despite occasionally almost-gloating over the adjustments in perception he is forcing on the Flatlanders, he refuses to even consider that his "universe" might be only part of a larger-dimensioned reality.
p. 187: when a series of observations follow a trend, there must be a reason.
p. 201: As part of the story-line, the protagonist tells a fairy-tale to his grandchildren at each New Year's celebration (only then?) which are all rehashes of Grimm's tales; only in the final one does he deviate - in order to explain the observed expansion of his own universe, he speculates that the Prince was able to find the Sleeping Beauty because the impenetrable forest expanded, increasing the distances between the trees enough for him to pass (which is very ingenious, I think, as the event is totally unexplained in Grimms). ( )