Challenges of the Late Sixteenth Century

The late sixteenth century posed numerous challenges, particularly for mathematicians. Amid political unrest, astronomers grappled with enormous numbers that made calculations painfully slow and prone to error. Multiplying and dividing these large figures could take months, and operations like square roots were equally tedious. A group of mathematicians, including our first key figure, sought to address these difficulties.

John Napier: A Pioneer of Calculation

John Napier, born in 1550, hailed from a noble background and had access to ample resources. His education remains somewhat of a mystery, but he ventured to mainland Europe for learning and returned fluent in Greek—an uncommon feat at the time. An astronomer himself, Napier aimed to tackle the computational challenges of his day. He invented a calculating device known as Napier's Bones, which facilitated multiplication and division. Despite this invention, his perfectionist nature drove him to seek even better methods.

The Swiss Innovator: Jost Bürgi

Jost Bürgi, born in 1552, was another brilliant mind of the era. As a clockmaker and mathematician, he created mechanical astronomical models and developed a table of sines vital for navigation. Although Bürgi kept much of his work private, collaboration with the renowned astronomer Johannes Kepler brought his contributions to light.

The Birth of Logarithms

While Napier continued his pursuit of efficient calculations, he encountered the concept of Prosthaphaeresis, which used trigonometry for rapid multiplication and division. His observation that powers of a number, when multiplied, formed an arithmetic sequence laid the groundwork for logarithms. In 1614, he published Mirifici Logarithmorum Canonis Descriptio, introducing the term 'logarithm' and presenting extensive logarithmic tables.

In parallel, Bürgi devised his own calculation method and kept it private until pressured by Kepler. In 1620, he published Progress Tabulen, revealing his independent invention of logarithmic tables, which are now recognized as antilogarithms.

Understanding Logarithms

At its core, a logarithm addresses the question: To what exponent must a base be raised to obtain a specific number? For instance, the logarithm of 64 with base 2 equals 6, meaning 2 raised to the power of x equals 64. This concept allows complex multiplications to be transformed into simpler additions using logarithm tables.

The English Mathematician: Henry Briggs

Henry Briggs, an English mathematician, encountered Napier's work while lecturing in London and proposed that logarithms be calculated using base 10. This suggestion simplified computations significantly and resonated with Napier, who embraced the idea. Although Napier passed away in 1617, Briggs further developed logarithmic concepts, leading to the discovery of the natural constant 'e' by Jacob Bernoulli.

Logarithms in Modern Times

Contrary to the belief that advancements in computational technology would render logarithms obsolete, they remain vital today. Logarithms have evolved into a fundamental mathematical function, serving as the inverse of exponential functions. They are integral in fields like population studies and pandemic modeling, helping to make sense of exponential changes in our world. Logarithmic scales, such as the Decibel and Richter scales, illustrate how we perceive phenomena that change exponentially.

The Importance of Understanding Logarithms

Given their rich history, it is essential for users of logarithms to appreciate the legacy behind them. While many students find logarithms mundane, they represent the hard work of visionary mathematicians striving to solve significant problems. As Pierre Simon Laplace remarked, Napier's logarithms "have doubled the life of the astronomer." We are indeed standing on the shoulders of giants, and it's crucial to acknowledge their contributions.

The first video titled "The History of the Natural Logarithm - How was it discovered?" provides a detailed account of the origins and significance of logarithms, featuring insights into their inventors and the mathematical concepts they developed.

The second video, "Logging the World - Oliver Johnson," explores the impact of logarithms on various scientific disciplines and their relevance in contemporary mathematics.

What Lies Ahead?

As we continue to utilize logarithms, understanding their historical context enriches our appreciation for this mathematical tool. Logarithms were not merely a subject in school; they were groundbreaking innovations that transformed the way we approach problems. The legacy of their inventors serves as a reminder of the ingenuity required to advance human knowledge.

References: Denis Roegel (LOCOMAT project on Bürgi's Progress Tabulen) and W.D. Cairns (scientific article on Napier's logarithms).