Why Newton’s second law is not F = ma

Abstract

The second law enunciated by Isaac Newton in the Principia is not equivalent to F = ma, as it is popularly known. The latter was described by Leonhard Euler, in 1752. However, for some historians, this formulation would be implicit in the statement proposed by Newton, and in this way, F = ma is considered by some of them only as a mathematical reformulation of Newton’s law. In this paper, initially, the considerations of these historians are used and is it is performed an interpretation in order to establish a connection between the law proposed by Newton and F = ma, using current mathematical language. Based on this interpretation and on a bibliographic research, made from primary and secondary historical sources, reasons for the non-equivalence of these laws are discussed, as well as aspects that lead to this confusion of interpretations; limitations of Newtonian mechanics are discussed too, which are about mathematical method, natural coordinates, and Newton’s conceptions of force. Next, the elements not covered by Newton’s mechanics are indicated, thus showing that, in fact, F = ma is a much more general law than that proposed by Newton. Finally, it is briefly discussed the importance of using the historical approach in Science Teaching, especially with respect to the episode discussed here.