The most appropriate method to obtain Euler’s F =ma (1776) from Principia’s Second Law (1686): A historical analysis

Abstract

Newton initiated physics, wrote various laws in the Principia in 1686, and explained phenomena with geometrical methods. Newton did not give any quantitative measure of force for Principia’s Second Law (PSL) even by geometrical methods, further did not write F =ma due to conceptual limitations. Euler gave F =ma in 1776 in various stages (F =ma/n in 1736, F =2ma in 1749,1752, F =ma/2 g in 1765). Then succeeding scientists tried to obtain Euler’s F =ma from PSL by two methods. Firstly, the definition of the second law is arbitrarily changed in standard textbooks, and the secondly change in motion was expressed in different ways. However, both existing methods are inconsistent. Thus, to obtain F =ma logically, the PSL is purposely altered (PSL-Altered) i.e. the rate of change of momentum is equal to the impressed force. This method has two advantages. Firstly, the definition of PSL is not randomly changed in textbooks; secondly, ‘change in motion’ is not assumed to have different values that are dimensionally inconsistent. As Euler’s F =ma is obtained from PSL, thus both Newton and Euler get due credits. Euler’s F =ma is logically obtained from Newton’s PSL-Altered; it indicates the far-reaching importance of PSL.