As a complementary example, we will consider the use of classical physics in a contemporary context, the derivation of the Rutherford scattering formula in a treatise on particle physics (Roe, 1996, pp. 19-25). We will focus more on the role of improper mathematics than on the details of the physics. Rutherford developed (or had a graduate assistant develop) the celebrated formula to explain the scattering of alpha particles through matter. Consider an incident particle with charge, ze, and momentum, p, with an impact parameter (perpendicular distance from the original path), b, relative to a target particle with charge, Ze. (Fig. 2). If Ze is the charge of an atomic nucleus and b is very small, then interaction with atomic electrons can be neglected. In this case the angle, , characterizing the deflection, may be calculated as the ratio of the perpendicular component of momentum, p, to the parallel component, p. p / p. From the definitions of force an the basis of electric force, it follows that

, using the relation dx = v dt. A little manipulation and the use of Gauss’s law leads to 2Zze2/bvp. The cross section for this happening at impact parameter, b, within a width, db, is d = 2 b db. The solid angle integrated over this cross section is d = 2 sin d. Dividing d by d yields a slightly the Rutherford scattering formula