The Histories.
by: Polybius
Polybius has said there is a cycle: Monarchy, the tyrant, then Aristocrate then oligarchy then democracy then what
According to Polybius’ theory of Anacyclosis, the next stage after democracy is Ochlocracy (mob rule). (AI - Google)
In this cyclical progression of government forms, Polybius identifies three "good" forms that inevitably degenerate into three "corrupt" counterparts:
Democracy (Rule by the Many – Just) is the stage where the people govern based on equality and collective voice.
Ochlocracy (Rule by the Many – Corrupt) occurs when democracy descends into chaos. In this stage, the masses are manipulated by demagogues, and the respect for laws and authority disappears, leading to social disintegration and "mob rule".
Monarchy (Cycle Restart): Out of the anarchy and violence of ochlocracy, a single strong leader eventually emerges to restore order, restarting the entire cycle back at monarchy.
The Full Cycle (Anacyclosis)
The predictable sequence of seven stages is as follows:
Monarchy (Original state/Kingship)
Tyranny (Corrupt rule by one)
Aristocracy (Rule by the best/virtuous few)
Oligarchy (Corrupt rule by the wealthy few)
Democracy (Rule by the many)
Ochlocracy (Mob rule/Corrupt democracy)
Monarchy (Cycle repeats)
Polybius argued that the Roman Republic was able to temporarily avoid this decay by adopting a mixed constitution that balanced elements of monarchy (consuls), aristocracy (the Senate), and democracy (popular assemblies).
can you tell me which part of his book that contains this cylce and which book
Polybius details this cycle, known as anacyclosis, in Book VI (Book 6) of his monumental work, The Histories. 
The specific discussion of the political cycle and its stages is found in the following sections of Book VI: 
Book VI, Chapters 3–4: Polybius introduces the six forms of government—three good (Kingship, Aristocracy, Democracy) and three degenerate (Tyranny, Oligarchy, Ochlocracy).
Book VI, Chapters 5–9: He explains the "natural" sequence or "rotation of polities" (anacyclosis). He describes how societies begin in a primitive state of nature, evolve into a monarchy, and then proceed through the cycle of growth and decay.
Book VI, Chapter 10: He concludes this section by explaining how lawmakers, such as Lycurgus of Sparta, recognized this inevitable cycle and designed "mixed constitutions" to avoid it. 
Context of the Book
In the broader scope of The Histories, which originally consisted of 40 books, Book VI serves as a famous philosophical "digression". Polybius uses it to explain why the Roman Republic was so successful: he argues that Rome’s government was a "mixed" system that combined elements of monarchy (the Consuls), aristocracy (the Senate), and democracy (the People), which temporarily allowed it to escape the cycle of decay. (Google - AI)
Books:
Delphi Complete Works of Polybius
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