The development of navigation in the 18^th Century was probably the most important driving force behind automated computation. It’s easy to tell how far north or south of the equator you are—you measure the height of the sun above the horizon at midday and then use the elevation and the date to work out your latitude. Calculating your longitude relative to the prime meridian through Greenwich in England is very much more difficult. Longitude is determined by comparing your local time, obtained by observing the angle of the sun, with the time at Greenwich; for example, if the local time is 8 am and your chronometer tells you that it’s 11 am in Greenwich, you are 360° x (11 - 8)/24 = 45° west of Greenwich. 

The mathematics of navigation uses trigonometry, which is concerned with the relationship between the sides and the angles of a triangle. In turn, trigonometry requires an accurate knowledge of the sine, cosine and tangent of an angle.  The sine of an angle can be calculated by hand. If x is expressed in radians (where 2p radians = 360°) and x < 1, the expression for sin(x) can be written as an infinite series of the form 

 Although the calculation of sin(x) requires the summation of an infinite number of terms, we can obtain a reasonably accurate approximation to sin(x) by adding just a handful of terms together because xⁿ tends towards zero as n increases for x <<1. 

Let’s test this formula. Suppose we wish to calculate the value of sin 15°. This angle corresponds to 15/2p radians = 0.2617993877991.

Step 1: sin(x) = x =  0.2617993878

Step 2: sin(x) = x – x³/3! = 0.2588088133

Step 3: sin(x) = x – x³/3! + x⁵/5! = 0.2588190618

The actual value of sin 15° is 0.2588190451 which differs from the calculated value only in the eighth decimal position.

When the tables of values for sin(x) were compiled many years ago, calculators were used to evaluate the successive terms in the series. These calculators weren't machines; they were armies of clerks who had to do the arithmetic the hard way—by means of pencil and paper. As you can imagine, people looked for a better method of compiling these tables.