Prologue

We owe to MINKOWSKI the fertile observation that certain results which can be made almost intuitive by the consideration of figures in n-dimensional euclidean space have far-reaching consequences in diverse branches of number theory. For example, he simplified the theory of units in algebraic number fields and both simplified and extended the theory of the approximation of irrational numbers by rational ones (Diophantine Approximation). This new branch of number theory, which MINKOWSKI christened “The Geometry of Numbers”, has developed into an independent branch of number-theory which, indeed, has many applications elsewhere but which is well worth studying for its own sake.