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A mechanical synthesizer with thirty harmonic elements (fifteen sine components and fifteen cosine components) may be used to graph a polynomial in a complex plane. The sum of the sine components is recorded by a tracing point which moves vertically. The sum of the cosine components is recorded by horizontal motion of a drawing board. Expansion of the polynomial by De Moivre's theorem expresses the function as a sum of sine terms and a sum of cosine terms. The polynomial may then be graphed by the machine, and thereby the complex roots determined. A number of typical curves are given to illustrate the machine method of determining real, imaginary, and complex roots. An auxiliary method is described which gives all the real roots of the polynomial from a single graph.

I. M. Vinogradov's method is used to derive estimates of partially rational trigonometric sums, independent of the degree of the rational part of the polynomial.

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(1968). Sine and Cosine Integrals. Optica Acta: International Journal of Optics: Vol. 15, No. 2, pp. 195-196.

Were it not for Euclid’s Propositions iii. 26 (proved by superposition) and vi. 33, the statement that he nowhere defines the circumference of a circle or any curved line would surely be a truism. His definition of a circle is substantially only a locus definition. It may well be argued that neither does he define a straight line (or an angle) ; but in the case of a straight line he introduces sufficient ideas and axioms to make his subsequent propositions convincing, whereas he gives us nothing to bridge the interval between a straight line and a curved one.And so, wise after the event, we may say that all the thought spent in old days over ‘squaring the circle’ had no adequate foundation, and was foredoomed to fail.As to iii. 26 and vi. 33, they are only further examples of the truth with which we are familiar in the case of angles, that it is sometimes easier to prove that things are equal etc., than to say exactly what they are.