The roots of polynomials of high degree are obtained with a stated accuracy in a time of ten to thirty minutes by use of an electromechanical harmonic synthesizer. The machine consists of sine‐cosine potentiometers which are driven by meshed gears to produce sinusoidal potentials of frequencyf, 2f, 3f, …. The algebraic polynomial is transformed into a trigonometric polynomial in which the terms are expressed in multiple angle form and applied to the machine. As the fundamental gear is rotated through 180°, the potentials are summed in a recording high impedance voltmeter in such a manner that when the curvef(&thgr;)=0 passes through zero values, the angles are read from a large dial and the rootsx=rcos &thgr; are obtained.Examples of the solutions are given, and the results of some twenty‐five solutions reported to show an average accuracy of 1.19% and an average time of solution of 16 minutes. For higher accuracy a time of about 7 minutes is required for each additional substitution, until an accuracy of 0.30% may be obtained. Multiple or repeated roots are obtained by use of the factor theorem and repeated differentiation.