On the deduction of Newton's second law

We learned at high school and university that Newton's second law is . However, Newton never wrote this equation. Furthermore, there is no agreement among historians of science as to the equation that expresses Newton's second law. On the other hand, Euler claimed to have discovered a principle of mechanics, which is . The respective paper of Euler provides us with the means of clarifying the issue. We can compare Newton's second law and Euler's principle with each other and verify whether there are significant differences between both laws. This is the task of the present paper. The result is that Newton's axiomatic system and technique of decomposing force differ from Euler's.