Having borrowed from the library an English translation of Newton's Principia (Motte's), I read the begining sections, Part 1 and the Systems of the world, and noticed that Newton did physics completely differently than is currently taught in a classical mechanics course. He uses synthetic geometry with certain theorems on limits, or a geometric calculus, which is very different from the symbolic calculus we used. Furthermore, he only uses scalar quantities instead of vector ones. Finally, he seems to define three types of forces, accelerative, absolute and motive forces. His derivation of Kepler's second law is completely different than the ones currently taught. When did physics texts then begin to discuss or teach newtonian mechanics in the modern analytical way using symbolic calculus, coordinate systems and vectors with free body diagrams?

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    $\begingroup$ Newton had literally created calculus for this work in physics, and therefore it was not widely known. He wrote his work in a geometric and more elementary style so that he could be more widely understood (and perhaps not accused of using sloppy mathematics). The use of vector notation came about hundreds of years later. Even Maxwell in the 19th century wrote his famous equations as lots of scalar equations instead of in a more compact form as 4 vector equations. The vector form of Maxwell's equations came 20 years later. See en.wikipedia.org/wiki/History_of_Maxwell%27s_equations. $\endgroup$
    – KCd
    May 14, 2015 at 0:22
  • $\begingroup$ Thanks for the input of vectors. What about the use of symbolic calculus, as Maxwell used, instead of Newton's geometric form? Also, what about using coordinate systems? Thanks again @KCd $\endgroup$
    – Cicero
    May 14, 2015 at 0:36
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    $\begingroup$ Leibniz had no qualms about using his convenient calculus notation, and the non-British mathematicians followed him. Since Newton and Leibniz were contemporaries, I think the Continental physicists started using symbolic calculus not long after calculus was developed. $\endgroup$
    – KCd
    May 14, 2015 at 0:55
  • $\begingroup$ Newton was writing for a specific audience. IIRC educated people of that time, even those we would today call "scientists," knew geometry but not algebra. $\endgroup$
    – user466
    May 15, 2015 at 23:04
  • $\begingroup$ @KCd, you might want to revise your first comment on Newton in light of the scholarship referenced in hsm.stackexchange.com/questions/6093/…. $\endgroup$ Jun 5, 2017 at 1:00

2 Answers 2


One cannot answer "when" because this was a "slow", gradual development. And it continues. Some milestones are the work of d'Alembert, Euler and Clairaut, then Lagrange's Mechanique analytique. It does not use a single figure, only formal calculus, and Lagrange was very proud of this. Then comes Laplace, Poisson, Hamilton and Hamiltonian mechanics. Vectors is a much later invention (end of 19 century, in physics). The modern formulation uses the language of manifolds, cotangent spaces and differential forms.

Modern high school and elementary university courses reflect approximately the level reached in the beginning of 20-s century, except vectors. Vectors penetrated elementary education only in the second half of 20-s century.

Remark. If you read what Newton wrote on Calculus, you will not recognize absolutely what is taught in modern courses. Modern calculus, as taught nowadays owes more to Leibniz, Bernoulli and Euler than to Newton. But here too there was a gradual development. Two parallel developments: on the advanced level, and another one, in elementary education which lags the advanced level by 50-100 years.


According to Truesdell [1954]:

(p. xliii:) As far as I can ascertain, it is Euler [1750, p. 196] which contains the first general statement of “Newton’s equations”. (p. xlii:) The axioms which Euler asserts “include all principles of mechanics” are $$ 2M\frac{d^2x}{dt^2}=P,\qquad 2M\frac{d^2y}{dt^2}=Q,\qquad 2M\frac{d^2z}{dt^2}=R. $$ (...) Anyone who has looked at Newton's Principia knows that no such equations occur in it.

I think he's right, except that Euler actually did it earlier in [1747, p. 103]:

Cela posé, prenant l'element du tems $dt$ pour constant, le changement instantané du mouvement du Corps sera exprimé par ces trois équations: $$ \textrm{I.}\quad \frac{2ddx}{dt^2}=\frac XM;\qquad \textrm{II.}\quad \frac{2ddy}{dt^2}=\frac YM;\qquad \textrm{III.}\quad \frac{2ddz}{dt^2}=\frac ZM $$ d'où l'on pourra tirer pour chaque tems ecoulé $t$ les valeurs $x$, $y$, $z$, & par conséquent l'endroit où le Corps se trouvera. C. Q. F. T.

Maltese [2003, 2006] has more on e.g. Euler and vectors.

  • [1747] Leonhard Euler. Recherches sur le mouvement des corps celestes en general. Hist. Acad. Roy. Berlin 3 (1749), 93–143. (“Presented on June 8, 1747”. Reprint: Opera Omnia (2) 25 (1960), 1–44.)

  • [1750] Leonhard Euler. Decouverte d’un nouveau principe de Mecanique. Hist. Acad. Roy. Berlin 6 (1752), 185–217. (“Presented on September 3, 1750”. Reprint: Opera Omnia (2) 5 (1957), 81–108.)

  • [1954] Clifford A. Truesdell. Rational fluid mechanics, 1687–1765. Leonhardi Euleri Opera Omnia (2) 12 (1954), ix–cxxv.

  • [2003] Giulio Maltese. The ancients’ inferno: the slow and tortuous development of ‘Newtonian’ principles of motion in the eighteenth century. In A. Becchi et al. (eds) Essays on the history of mechanics, Birkhäuser, Basel, pp. 199–221.

  • [2006] Giulio Maltese. On the changing fortune of the Newtonian tradition in mechanics. In K. Williams (ed.) Two cultures, Birkhäuser, Basel, pp. 97–113.

  • $\begingroup$ My question really was asking for this kind of physics, so I think that Euler is who I was looking for. $\endgroup$
    – Cicero
    May 14, 2015 at 2:56
  • $\begingroup$ In this essay Truesdell attributes the law of conservation of angular momentum to the Bernoullis (James and Daniel) and Euler: books.google.de/… $\endgroup$ Feb 27, 2017 at 13:58

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