A short lecture and Q&A with John Butcher
In the early days of Runge-Kutta methods, 1895-1901, the numerical order of particular methods was found by working out the Taylor series for the solution of a generic scalar differential equation. By comparing this with the Taylor series for the Runge-Kutta result, the order of the approximation can be found. This approach was highly successful in finding methods up to order 4 but it became more and more complicated until eventually, in the 1950s, methods up to order 6 were found.
It was discovered, about that time, that the theory based on a scalar problem was not adequate, and a vector differential equation system had to be used as the test problem.
The theory, arising from this high-dimensional approach, has now become B-series analysis.