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Chapter 4. The Rainforest Library.
Mr. Ruche receives the crate of books that Grosrouvre sent him. The collection is called the Rainforest Library!
Mr. Ruche, Perrette, the twins, and Max discuss together how to classify the "Rainforest Library." They agree to organize the crate of books into the next categories: Geometry, Arithmetic, Algebra, Trigonometry, Probability, Mechanics, Logic, and the New Maths!
Max visits a pet shop along with Sidney; he is looking for food for the parrot, there, he sees an official notice stating that to have a pet in France this one needs to have an official certificate.
He is told by a girl working there that Sidney is a valuable bird talker, an Amazon blue. The boy has a bad feeling with the girl and quickly gets out of the pet shop. After that, the girl calls someone giving details about Max and Sidney!
Chapter 5. The Three Ages of Math.
Mr. Ruche visits the library for several days, where he studies who were the most important mathematicians and their contributions in different periods of the history.
The first day he compiles the list of mathematicians from Ancient Greece, from 700 BC to AD 700. The second day, he continues with the distinguished names in the Arab world, from AD 800 to 1400. And finally, he finishes listing the people of the Western world, from 1500 to 1900.
After that, he along with Perrette, start to classify the books in the crates, one by one. And he realizes that, indeed, Grosrouvre was not exaggerating when he said in the letter that was the world's best collection of books about mathematics.
Chapter 6. Friends and Enemies
Mr. Ruche receives an envelope from Brazil that informs him that Grosrouvre was found dead; another letter is inside the pocket, it is directed to him and written by Elgar.
Perrette reads the letter to Mr. Ruche. In the letter, Elgar tells him that he solved two theorems and decided to keep them in secret. However, some people with bad intentions was trying to know the solution of those theorems and were behind him. He says goodbye.
Mr. Ruche is sad because of his best friend.
He, Perrette and the boys (except Lea) discuss the content of the letter, and they try to work out what happened with Elgar.
Chapter 7. The Numbers Game
Mr. Ruche examines the life of Pythagoras and his disciples, the so-called Pythagoreans, such as Hippasus, the leader of the initiated or acousmaticians; Hippocrates, who studied the crescents or lunes and was thrown out of the Pythagorean school; Philolaus, the survivor of the fire provoked by a rejected aspirant, Cylon. Philolaus was an astronomer. Also Archytas, who was an engineer and is reputed to have made a mechanical bird, Archytas also saved Plato from a tyrant called Denys.
After reading about Pythagoras, he along with Max and Sidney prepares a lecture to explain to the rest other details about Pythagoras, such as the relationship between musical notes and numbers, the Pythagoras's theorem, and how the initiation was done in his school.
The aspirants to be his student were supposed to wait for five years in silence behind a curtain. Those who were waiting to see the master face to face were called exoterics, and the ones who achieved the honor were known as esoterics.
Mr. Ruche says to the guys that they are worthy of being esoterics. Mr. Ruche discusses with Perrette about Grosrouvre. She thinks that they should consider the possibility that he is not dead.
Finally, and because of Lea was insisting that she was starving, Mr. Ruche suggests to go for an all-nighter and Albert, who was there all the time, wakes up and refuse to be an esoteric.
Chapter 8. Just a Fraction
Mr. Ruche organizes a night school where he discusses the root of number two.
Perrette reflects on all the things that are happening since they received the letter. Max is pouring the water from one vase into another, to help Sidney to reach the water in the glass; she quickly realizes that the water will overflow because she sums the ratio of water in each vase, Max gets impressed.
Albert tells Max about the knowledge he has acquired of different cities in his profession as a taxi driver, even when he has never been in those cities.
The twins together work at the night and proof that there is not a fraction, that, squared, could equal two. Perrette and Mr. Ruche are impressed; he approves his evolution from acousmaticians to mathematicians.