math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots. Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations. It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation, [1] where the sign of the ± symbol is chosen so that and may both be positive. By substituting [2] and then multiplying through by , we obtain [3] Introducing functions of and rearranging, we obtain [4] [5] where the subscripts and correspond, respectively, to the use of a negative or positive sign in equation [1]. Substituting the two values of or found from equations [4] or [5] into [2] gives the required roots of [1]. Complex roots occur in the solution based on equation [5] if the absolute value of exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone. Calculating complex roots would require using a different trigonometric form. To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy: A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries. (rounded to six significant figures) Solution for complex roots in polar coordinates If the quadratic equation with real coefficients has two complex roots—the case where requiring a and c to have the same sign as each other—then the solutions for the roots can be expressed in polar form as where and Geometric solution The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients , , are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from