Logarithm and Logarithm Table
Introduction
Logarithm tables are commonly listed in the appendices of Mathematics textbooks to ease readers’ calculations. However, do you know it was before the invention of the concept of logarithm that the first logarithm table was published?
Background
In the fifteenth century, the Renaissance in Europe sped up the development of distant transport. Yet, at that time, it was a tedious work to perform multiplications and divisions on the large numbers involved in distant transport. And there arouse a question: Does there exist an accurate and efficient way to perform arithmetic operations on large numbers?
Replacing Multiplication and Division with Addition and Subtraction
In order to simplify the calculations, a natural thought was to replace multiplication and division with addition and subtraction.
This idea was illustrated by two transformation formulae emerging in the sixteenth century:
However, both formulae were not the perfect answers to the question: the former required manipulations of sine and cosine ratios in trigonometry, while the latter needed evaluations of squares of the numbers, and this would involve some tedious multiplications.
A Discovery
In 1484, Nicolas Chuquet, a bachelor of medicine in Paris, discovered an interesting mathematical phenomenon:
2 + 4 = 6
The first row is an arithmetic sequence with a common difference of 1, and the second row is a geometric sequence with a common ratio of 2.
He discovered that the product of any two terms in the geometric sequence lies in the sequence, and the sum of the two corresponding terms in the arithmetic sequence corresponds to this product too.
For example, consider the terms 4 and 16 in the geometric sequence. Their product, 64, is a term in the same sequence. Besides, as the number above 4 ´ 16 = 64 is 6, this product corresponds to the number 6, which is the sum of the two terms (2 and 4) corresponding to 4 and 16 in the sequence.
Chuquet’s discovery was inspiring. The implication was that one can convert multiplication to addition by a correspondence between an arithmetic sequence and a geometric sequence.
The Embryo of the Logarithm Table
Half a century later, a German Mathematician named Michael Stifel introduced the use of negative numbers and rational numbers in geometric and arithmetic sequences. He went further to point out that multiplication and division in geometric sequences can be converted to addition and subtraction in arithmetic sequences respectively.
For example, to evaluate , one may notice that 5 is the number corresponding to 32 in the arithmetic sequence and -1 is that corresponding to . As 5 – (-1) = 6 and 6 corresponds to 64, the answer is 64.
5 – (-1) = 6
The Invention of the Logarithm Table
The first person to give a definition to logarithm and propose the logarithm table is John Napier, a Scottish mathematician. He defined logarithm with the help of physical motions.
He assumed that there are two particles P and Q traveling along the rays AZ and A’Z’ respectively. P maintains a constant velocity, while the velocity of Q at any point is proportional to the remaining distance (this means that the velocity of Q is decreasing).
Suppose that when P gets to B’, Q is at B, and when P gets to C’, D’, E’, …, Q is at C, D, E, ….Then, the distance A’B’ is equal to the logarithm of the distance BZ. Similarly, distances A’C’, A’D’ and A’E’ are equal to the logarithms of the distances CZ, DZ and EZ respectively.
Q: | |
P: |
In this way, Napier defined the logarithmic values of the numbers in a geometric sequence to be the corresponding numbers in the corresponding arithmetic sequence. Hence, the logarithmic values defined by Napier with physical motions or geometry are continuous, while those defined with sequences are discrete. Of course, even Napier could not list out the logarithm of every number in the sequence when he constructed his logarithm table.
Napier spent nearly 20 years to construct his logarithm table to a precision of 7 significant figures.
A More Precise Logarithm Table
The publication of Napier’s logarithm table was fully supported by the British Mathematician Henry Briggs, who then suggested that Napier should use base 10 in his logarithm table for the sake of convenience in the decimal system.
While Napier agreed to Briggs’s advice, he was too tired to finish off the work. To accomplish the goal, Briggs and a young Dutch mathematician Adriaan Vlacq constructed the common logarithm table, to the precision of 14 significant figures, for the numbers between 1 and 100000.