  In book II of Newton's "Principia Mathematica" of 1687 several applicative
problems are introduced and solved. There, we can find the formulation of the
first calculus of variations problem that leads to the first free boundary
problem of history. The general calculus of variations problem is concerned
with the optimal shape design for the motion of projectiles subject to air
resistance. Here, for Newton's optimal nose cone free boundary problem, we
define a non-iterative initial value method which is referred in literature as
transformation method. To define this method we apply invariance properties of
Newton's free boundary problem under a scaling group of point transformations.
Finally, we compare our non-iterative numerical results with those available in
literature and obtained via an iterative shooting method. We emphasize that our
non-iterative method is faster than shooting or collocation methods and does
not need any preliminary computation to test the target function as the
iterative method or even provide any initial iterate. Moreover, applying
Buckingham Pi-Theorem we get the functional relation between the unknown free
boundary and the nose cone radius and height.
