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Analogue and digital computing machines are both capable of solving numerous types of mathematical problems. Nevertheless, these two kinds of machines mechanize the solutions in fundamentally different ways. The problem of simulation in real time is used as a basis for comparing the characteristics of the two kinds of machines as regards basic units, size, speed, precision, accuracy, stability, input-output, flexibility and future possibilities. The mathematical operations discussed include addition, subtraction, multiplication, division, evaluation of polynomials and other functions, square root extraction, and the solution of ordinary differential equations.