Starting with known special cases, the calculation of logarithms and trigonometric functions can be performed by looking up numbers in a mathematical table, and interpolating between known cases. For small enough differences, this linear operation was accurate enough for use in navigation and astronomy in the Age of Exploration. The uses of interpolation have thrived in the past 500 years: by the twentieth century Leslie Comrie and W.J. Eckert systematized the use of interpolation in tables of numbers for punch card calculation.
In our time, even a student can simulate the motion of the planets, an N-body differential equation, using the concepts of numerical approximation, a feat which even Isaac Newton could admire, given his struggles with the motion of the Moon.
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