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Let $K$ be a commutative field and let $V$ be a subspace of $P^{n}(K)$. Let$\Gamma$ be a subgroup of $\operatorname{Aut}(K)$ and let $\Gamma$ act on$P^{n}(K)$ by $\sigma((x_{i})_{0\leq i\leq n})=(\sigma(x_{i}))_{0\leq i\leq n}$for $\sigma\in\Gamma$ and $(x_{i})_{0\leq i\leq n}\in P^{n}(K)$. In this paper,we ask `how much' unstable $V$ is under $\Gamma$ by asking how much higher orlower dimension the join and the meet of $\sigma(V)$ ($\sigma \in\Gamma$) havethan $V$, and answer it in terms of the Pl\"{u}cker coordinates of $V$ and theinvariant field $k$ of $\Gamma$, through presenting two `irrationality'measures of $V$ over $k$.

Two instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$