  We consider the error analysis of Lagrange interpolation on triangles and
tetrahedrons. For Lagrange interpolation of order one, Babu\v{s}ka and Aziz
showed that squeezing a right isosceles triangle perpendicularly does not
deteriorate the optimal approximation order. We extend their technique and
result to higher-order Lagrange interpolation on both triangles and
tetrahedrons. To this end, we make use of difference quotients of functions
with two or three variables. Then, the error estimates on squeezed triangles
and tetrahedrons are proved by a method that is a straightforward extension of
the original given by Babu\v{s}ka-Aziz.
