  This is an annotated translation of E126 'De novo genere oscillationum', in
which Euler derived for the first time, the differential equation of the
(undamped) simple harmonic oscillator under harmonic excitation, namely, the
motion of an object subjected to two acting forces, one proportional to the
distance travelled, the other one varying sinusoidally with time. He then
developed a general solution, making extensive use direct and inverse sine and
cosine functions. After much manipulation of the resulting equations, he
proceeds to analyze the periodicity of the solutions by varying the values of
the parameters, to finally find out the phenomenon of resonance by saying "...
Among all these cases, the one which deserves particular attention is that for
which 2b=a, in which the oscillation distance eventually grows up to infinite:
this effect is most remarkable, since it is generated by finite forces."

History and Philosophy of Physics