Part 3 of the introduction starts with a few pages about methodology. The short version is that the best read on incomes comes from income tax data, and on wealth from estate tax data. Most modern industrialized countries have had income taxes for about a century, and estate taxes for longer than that. France in particular has kept wealth and estate tax data since the 1789 Revolution. This data, as well as other data on incomes, are kept in publicly-accessible databases. Thomas Piketty concludes the section with a bit about how much easier it is to obtain and process large amounts of data with the benefit of computers and the internet—something his predecessors lacked.
Next, the main conclusions of the book: One is that the reduction of inequality between 1910 and 1950 was a direct effect of two world wars and policies adopted to cope with them. The other conclusion is that “there are powerful mechanisms pushing alternatively toward convergence (less inequality) and divergence (more inequality). Furthermore, there is no natural, spontaneous process to prevent destabilizing, inegalitarian forces from prevailing permanently.”
We see a great example of this in “The Rich Don’t Always Win,” by Sam Pizzigati, in which FDR is unable to do very much to reduce inequality, even with a heavily Democratic congress, but then the war gives him the ability to raise taxes to a much higher level as well as freeze wages for higher-earners while allowing lower-earners’ wages to creep upward. The second conclusion is illustrated nicely in the BBC “Origin of Debt” series, in which the speaker mentions frequent “jubilee” years in the Ancient World, in which debts would be cleared.
Thomas Piketty mentions several potential forces for convergence, such as the diffusion of skills. (Witness students from poorer countries learning the latest science from American and European universities and bringing those skills back home.) He also mentions that, in theory, “human capital” (individual knowledge and skills) could play a progressively dominant role, pushing wages up, but notes that this doesn’t necessarily cause convergence, as it could just result in certain people whose skills are highly in demand making exorbitant salaries.. Another mechanism cited is “generational warfare” in which people of all income levels save, then spend (and benefit from social programs) when old and retired. However, he dismisses these two theories as, while good-sounding in theory, simply not the case in practice, given the large increase in inequality we have seen over the last several decades, and that labor’s share of total income has been very stable for over a century.
Two forces of divergence are cited. 
One, most prevalent in the United States (but present elsewhere) is the increase in super-high incomes relative to other incomes. (See the chart to the right.)
The other force is the primary focus of the book: r > g, where r is the (inflation-adjusted) rate of return for capital, and g is the growth of the economy in general. Under this inequality, those with wealth can spend a piece of their income, reinvest the rest, and end up wealthier, resulting in the wealthy getting a larger share of total income with each generation. The table above, starting after 1950, illustrates this nicely (thought it looks at wealth rather than income), as does this Tom the Dancing Bug comic.
Other forces may compound this, such as wealthier people saving a larger % of their income, and/or getting a higher rate of return than less wealthy people. (We see a lot of this. As a minor example, I personally have dealt with fund managers and mutual funds that charge a smaller % as you invest more.)
I read Capital in the Twenty First Century in 2014-2015 and was extremely impressed by both the content and the presentation. In my views, it’s easily in the running for most important economics book of the last few . This year I am doing an in-depth, chapter-by-chapter review of the book. One chapter will be reviewed and discussed every two weeks.