  A Latin square of side n defines in a natural way a finite geometry on 3n
points, with three lines of size n and n^2 lines of size 3. A Latin square of
side n with a transversal similarly defines a finite geometry on 3n+1 points,
with three lines of size n, n^2-n lines of size 3, and n concurrent lines of
size 4. A collection of k mutually orthogonal Latin squares defines a geometry
on kn points, with k lines of size n and n^2 lines of size k. Extending work of
Bruen and Colbourn (J. Combin. Th. Ser. A 92 (2000), 88-94), we characterise
embeddings of these finite geometries into projective spaces over skew fields.
