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# Isaac Newton, Leonhard Euler and F = ma

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In Newton's time, the terms acceleration and second derivative did not exist, so he could not have written F = ma but unscientifically credited to Newton. Euler directly and independently gave various equations of force F = ma/2 or F = ma/n in 1736, F = 2ma in 1750 (the coefficient 2 depended on the unity of measure), F = ma/2g in 1765 and F = ma in 1775 in different ways. Further, F = ma is used to derive the rest mass energy equation (E 0 = m 0 c 2), but if F = 2ma, F = ma/2, and F = ma/2g are used then equations for rest mass energy become E 0 = 2m 0 c 2, E 0 = m 0 c 2/2, and E 0 = m 0 c 2/2g. The situation is similar to kinetic energy. The equation for Principia's Law II is F = k(v−u), which is not discussed at all. Newton gave a velocity dependent force and Euler an acceleration dependent force. If the magnitudes of equations of force F = ma and F = 2ma are equal then it implies 1 = 2, which is not true. Euler used two primary or fundamental units L (length) and F (force), hence the coefficient is 2. The systems of primary units L (length), F (force), T (time) and L (length), m (mass), T (time) were introduced in the following century. Thus equation F = ma may be understood as a postulate. The terms “motion” and “quantity of motion,” are required to be properly understood as defined in the Principia. Further the terms “alteration” and “rate of change of” are entirely different. In highly specialized literature it is mentioned that Euler invented F = ma (48 years after death of Newton), but in the standard literature or textbooks level physics Newton is the originator of F = ma. The debate concerning various aspects of F = ma is wide open and will reveal new facts in science.
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Keywords: Euler; F = ma; Newton

Document Type: Research Article

Affiliations: Fundamental Physics Society, His Mercy Enclave, Post Box 107 GPO, Shimla 171001, HP, India

Publication date: September 16, 2014

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