Let q(x1; …, xn) be a positive definite quadratic form in n variables with real coefficients. Minkowski defined the successive minima of q as follows. Let S1 denote the least value assumed by q for integers x1 …, xn, not all zero, and let
be a point at which this value is attained. Let S2 denote the least value assumed by q at integral points which are not multiples of x(1), and let x(2) be such a point at which this value is attained. Let S3 be the least value of q at integral points which are not linearly dependent on x(1) and x(2), and so on. We have
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300000486/resource/name/S0025579300000486_eqnU1.gif?pub-status=live)
and it is easy to see that these numbers are uniquely defined, even though there may be several choices for the points x(1), …, x(n). The determinant N of the coordinates of the points x(1), …, x(n) is a non-zero integer. We denote by N (q) the least value of this integer (taken positively) for all permissible choices of the n minimal points, and by N′(q) its greatest value. Plainly N(q) and N′(q) are arithmetical invariants of q, that is, they are the same for two forms which are equivalent under a linear substitution with integral coefficients and determinant ±1.