LogarithmsDuring the sixteenth century, several people were working on using series of powers of numbers (for example, 1, 2, 4, 8, 16, 32, 64...) to simplify solutions of various mathematical problems. It was noted that relations such as:
(xn)(xm) = xn+m
(xn) / (xm) = xn-m
held for any values of x, n and m, and that this might be useful in simplifying complex equations involving products. John Napier published a book in 1614 explaining how to define a logarithm in order to help with the multiplication of large numbers. Napier's example was that of a triangle with a hypotenuse of 107 units, the kind of order of calculation needed in astronomy.
A number, n, is the logarithm of x to a given base b if and only if the following is true:
x = bn The base, b, can be any number but Napier chose a value called e = 2.7182818, which simplified some of the calculations he was working on.
Using the equalities above, formulas such as those below can be easily derived and used to simplify calculations:
logb (mn) = logb m + logb n
logb (m/n) = logb m - logb n
logb (np) = p logb n
For example, the law of sines for spherical triangles becomes:
Log (sin a) - log (sin A) = log (sin b) - log (sin B) = log (sin c) - log (sin C)
Which is much simpler to calculate than:
(sin a)/(sin A) = (sin b)/(sin B) = (sin c)/(sin C)
The decrease in errors due to the use of logarithms was not just because of the ease of calculation, but also because the values in the tables of sines and cosines were approximations to the actual values. If, for example, a measurement of distance was being made between two objects 107 units away (Napier's example), the angle between their positions would be small and so the sine of the angle would approach 0, and the cosine would approach 1. Even accurate multiplication of these values would compound errors in the trigonometric tables being used (errors in the tables are greater near 0 and 1 because of the behaviour of the sine and cosine functions). The use of logarithms allowed errors to be minimised, at least as much as the logarithm tables were accurate, and improved calculations.
Kepler's Rudolphine Tables, published in 1627, made use of the new logarithmic techniques, giving far more accurate values for the latitudes of stars than previously. Combining logarithms and trigonometry led to the development of a new set of relations between the sides and angles of a spherical triangle including Napier's Analogies. These are formulas relating the half angles and half sides and are well suited to logarithmic solutions of a spherical triangle.
The swift development of logarithmic methods for the solution of spherical triangles, and the number of publications of tables and procedures for the calculation of logarithms attest to the immediate contemporary importance of the new technique. Calculating machines were quickly designed, including those of Strickard (at the University of Tübingen) and the famous Napier's Bones. These calculating machines worked by having logarithmic scales that enabled multiplication to be quickly and easily reckoned without the need for long-multiplication.