Lie's group theory of differential equations, which is more than a hundred years old, has had a rebirth in
the last twenty years as people have again begun to appreciate its virtues, namely: A) that it unifies the
many ad hoc methods known for solving differential equations, and B) that it provides powerful new
ways to find solutions. The theory has applications to both ordinary and partial differential equations and
is not restricted to linear equations. What is needed now is a short, simple, introductory text to spread the
word.
I am not satisfied that recent books fill this need. Like many modern mathematics books, they are
written in an encyclopedic style that makes them hard to read. In striving for rigor and generality right
from the outset, their authors make the underlying ideas not only difficult to understand but sometimes
even difficult to identify. The book that we want should be, to borrow a phrase of Jacques Barzun's,
'simple and direct.'