" Every curve of the n-th order is in a flat space of n dimensions or less " 


and


" If there be a system of $n + m + 1$ quantities $x$ connected by $n + m - 1$ homogeneous equations ; and if this system be such that, upon the addition to it of one equation more, linear and homogeneous in the quantities os and having arbitrary coefficients, it gives $n$ sets of values for the ratios of the quantities $x$, these $n + m + 1$ quantities can always be expressed as linear and homogeneous functions of $n + 1$ quantities." 


(quoted from *Mathematical Papers*, 1882, W. K. Clifford, p. lxviii)



Why these two propositions are equivalent  ?
In modern terms, a curve of degree $n$ in some projective space of dimension $n+m$ is actually contained in a projective subspace of dimension $n$. For a proof, project from a point on the curve to a hyperplane and argue by induction on $n$. The second paragraph is just recasting the definition of curve of degree $n$ as one that has $n$ points of intersection with a generic hyperplane.


