A translation of L. Euler's "On a new class of oscillations"

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."