Logarithms: why do they even exist?

The History of Logarithms

Key Concepts: Logarithms, John Napier, Yoast Burgi, Henry Briggs, prosthaphaeresis, geometric series, arithmetic sequence, base 10 logarithms, common logarithm, exponential function, Weber's law.

Introduction

The video explores the history of logarithms, aiming to foster an appreciation for their significance in mathematical history. It introduces the key figures involved in their development and their impact on science and mathematics.

The Historical Context

The video sets the scene in the late 16th and early 17th century, highlighting the challenges faced by mathematicians and astronomers dealing with large numbers. The need for faster multiplication, division, and root extraction methods was becoming increasingly important due to advancements in astronomy and the use of telescopes.

The Main Players

• John Napier: A Scottish laird known for his work in computational mathematics, including Napier's bones. He sought a more efficient method than prosthaphaeresis for simplifying calculations. In 1614, he published A Description of the Wonderful Canon of Logarithms, detailing his work on logarithms after 20 years of development. He aimed to alleviate the "tedium of lengthy multiplications and divisions" and the "annoyance of the many slippery errors that can arise."
• Yoast Burgi: A Swiss clock maker and mathematician who independently invented his own version of logarithms around the same time as Napier. He delayed publication until 1620, prompted by Johann Kepler.
• Henry Briggs: A professor of geometry at Oxford who greatly admired Napier's work. He arranged a meeting with Napier to discuss potential improvements to the logarithms.

The Development of Logarithms

Napier and Burgi both used the same fundamental principles, but approached it slightly differently. The video explains the concept of logarithms using the example of powers of two.

• Geometric Series and Arithmetic Sequence: The powers of two (2, 4, 8, 16...) form a geometric series, while their corresponding term numbers (1, 2, 3, 4...) form an arithmetic sequence. Multiplying numbers in the geometric series corresponds to adding their term numbers in the arithmetic sequence.
• Logarithms as Term Numbers: Logarithms are essentially the term numbers in this context. For example, the base 2 logarithm of 8 is 3, because 2 raised to the power of 3 equals 8.
• Handling Non-Powers of Two: Napier and Burgi used powers of numbers very close to one (slightly less than one for Napier, slightly greater than one for Burgi) to create tables that covered a wide range of numbers with high accuracy. Linear interpolation could be used to approximate logarithms of numbers between the terms in the table.

The Napier-Briggs Collaboration

In their meeting, Briggs suggested making the logarithm of 10 equal to 1, effectively creating base 10 logarithms. This was crucial because it allowed for the creation of more manageable log tables.

• Base 10 Logarithms: With base 10 logarithms, the logarithm of a number like 456 can be expressed as the logarithm of 4.56 plus the logarithm of 100. This meant that log tables only needed to cover numbers up to 10, as the magnitudes of 10 could be easily added.
• Briggs's Contribution: After Napier's death, Briggs took on the work and created new log tables of base 10, which became the common logarithms used today.

The Impact and Modern Relevance of Logarithms

Logarithms significantly accelerated the development of mathematics, engineering, and science.

• Praise from Mathematicians: Johann Kepler dedicated his next book publication to Napier, and Pierre-Simon Laplace stated that "Logarithms, by shortening the labors, has doubled the life of an astronomer."
• Modern Applications: While computers and calculators have rendered logarithm tables obsolete for their original purpose, the concept of logarithms remains crucial. Logarithms are now understood as the inverse of the exponential function and are used to analyze and manipulate exponential functions.
• Examples: Logarithms are used to calculate the cooling rate of coffee (exponential decay) and are relevant to Weber's law, which describes how humans perceive stimuli logarithmically.

Weber's Law

Weber's law states that we perceive changes in stimuli in proportion to the amount of the stimuli that was already there. The video uses the example of dots to illustrate this concept. The same change in the number of dots is more noticeable when there are fewer dots to begin with.

Conclusion

The video concludes by emphasizing the enduring importance of logarithms, both historically and in modern applications. While their original purpose has been superseded by technology, the underlying concept remains fundamental to mathematics and our understanding of the world.