Contributed by
Garmt de Vries
The members of the Gun Club want to use a giant cannon's recoil to change the Earth's rotation axis, so they can exploit the presumed coalfields at the North Pole. An unfortunate side effect is that many regions will be submerged in the sea or end up at unpleasant altitudes. The world holds its breath as the gun is fired, but nothing happens. It turns out that Maston, who did all the calculations, forgot three zeros in the beginning, which led to a result that was wrong by a factor 10^{18}.
The novel contains a supplementary chapter by Alain Badoureau, a mine engineer who helped Verne with all the math. In this extra chapter, all the calculations are rigorously presented.

Garmt de Vries has written to me with other suggestions of works by Jules Verne to add to this database. Some of them fall below the borderline of what I consider to be mathematical fiction. (I'm afraid that I have to be somewhat restrictive about what is included, or else the list will become too long to be useful!) Sans Dessus Dessous is included not because of the appendix containing computations, but because mathematics is truly essential to the plot and because it says something about mathematics.
Here are some other suggestions from de Vries:
Contributed by
Garmt de Vries
 Voyage au centre de la Terre (1864, English: Journey to the centre of the Earth) starts with the deciphering of a cryptogram, using a simple transposition. See chapters 25.
 Mathias Sandorf (1885) also starts out with a message encrypted by transposition, but slightly more complicated, with the use of a grid. See vol. 1, ch. 4.
 La jangada (1881, English: Eight Hundred Leagues on the Amazon) is all about cryptography. The first lines of the novel consist of the cryptogram to be solved. In the end, a man's life depends on the information in this document, and several chapters are spent trying to decipher it. The system used is a Vigenere system, where each letter of the clear text can be represented by different letters in the cryptotext, depending on its position. The solution is found in the nick of time, when the name of the document's author is found out. This yields the key to the cryptogram, and the protagonist's life is saved. When this novel was first serialised before appearing as a volume, a math student apparently solved the riddle before the necessary clue was given or even hinted at. Jules Verne went to see the student, who explained how he had done it. Verne was much impressed. The method used by the student is not known. It may have been something like Kasiski counting, or using a probable word approach.
 Aventures de trois Russes et de trois Anglais (1872, English: Meridiana) is about an AngloRussian team of astronomers who set out to measure a meridian in southern Africa. Chapter 4 gives all the gory details of trigonometry, along with a history of the metre, and why it is important to measure a meridian. One of the characters is an absentminded mathematician, who wanders off into the wild for days and is almost eaten by crocodiles, because he is verifying the logarithmic tables of James Wolston (ch. 9).
 In Hector Servadac (1877, English: Off on a comet), a group of people is taken away by a comet that collides with the Earth. One of them is a French astronomer, who is often compleely absorbed in his calculations. At one point, he gives a lecture on how to calculate the mass, density, etc. of their comet. See vol.2 , ch. 8.
 Mirifiques aventures de MaÃ®tre Antifer (1894) is based on a geometrical problem. Three characters inherit an immense treasure, which is buried on an island. Each time they receive a longitude that they have to combine with a latitude in the possession of another heir to find the location of a new island. In the end, they have visited three islands, and find a document that has become illegible. There are some traces of text: "it suffices... circumference... pole...". The location of the last island is the centre of the circle through all three islands. It is a bit silly that they determine this final location using only a globe and a ruler, but the idea is nice.

Contributed by
Vijay Fafat
In this section, you've listed many works by Verne involving Math. I think the following two also belong to that list:
a. From the Earth to the Moon: where the author discusses details of possible geometric ways of communication with extraterrestrials, mathematical considerations of the trip to the moon, orbital recalculations, etc
b. Mysterious Island: where geometry and a simple apparatus is used to fix the castaway's longitude and latitude.

