John Napier: The Revolutionary Mind Behind Logarithms

John Napier went to the University of St. Andrews at 13 but did not graduate. Instead, he traveled in Europe, returned to Scotland, managed his father’s land, got married twice, and explored different things like religion, farming, and astronomy. His big achievement was creating logarithms, which he started working on in 1594. In 1614, he published a key work called “Mirifici logarithmorum canonis descriptio,” where he introduced the term “logarithm” and changed mathematics. Logarithms were particularly important in the days before calculators, as they allowed lengthy multiplications and divisions to be performed by addition and subtraction of the “logarithms” of the original numbers. 

Early Life and Education  

Napier was born on February 1, 1550, in Merchiston Castle near Edinburgh (see below), which is now a cherished part of Edinburgh Napier University now. 

Remarkably, John’s journey into academia commenced at an early age, beginning his scholarly path at the age of 13, this may seem astonishing now but at the time this was the normal age most students went to university. He matriculated in St Salvator’s College in 1563, and his mother arranged for him to live in the College under the supervision of its Principal, John Rutherford. However, the early days of John Napier’s academic journey were marked by tragedy, as he lost his mother shortly after enrolling.  

Yet, Napier’s journey at the university was short-lived, as he chose to leave before earning a formal degree. Surprisingly, it was not at St. Andrews that he gained his profound insights into higher mathematics or classical literature. Instead, these intellectual treasures were garnered during his studies abroad in Europe.

Image shows a statue set into an arch in a stone wall with stone decorative elements above.
Statue of John Napier.
Image credit: Stephencdickson, CC BY-SA 4.0 DEED via Wikimedia Commons
image shows a picture of a grand stone building with a small tower on the left side, surrounded by bushes, with trees in the foreground
Photo of Merchiston Tower, John Napier’s birthplace.
Image credit: public domain via Wikimedia Commons.

Return to Scotland  

In 1571, John Napier returned to Scotland, married Elizabeth Stirling, and had two children. Unfortunately, Elizabeth passed away in 1579. Later, he married Agnes Chisholm, and they had 10 children. His father gave him land on which he built a castle in 1574, and he inherited the rest of the Merchiston estates on his father’s death in 1608. 

Despite his noble status exempting him from a traditional profession, he pursued various interests. His deep religious commitment led him to participate in religious matters in Scotland. As a steward of his estate, Napier worked on enhancing its operations, including the development of fertilizers to improve crop yields and the creation of devices to combat flooding. He also delved into land surveying, measurement, and devised military strategies and tools to safeguard his land and nation. 

Astronomy ignited his passion and led him to explore the captivating realm of celestial wonders. His research and complex calculations involving large numbers sparked his creativity, leading to one of mathematics’ most enduring inventions: the logarithm. John Napier’s thirst for knowledge knew no bounds, leading him to pen several books that would immortalize his contributions to mathematics. His remarkable journey reached its end on April 4, 1617, as he succumbed to complications from gout, leaving behind a legacy that continues to shine brightly.  

Invention of Logarithms  

Napier’s main occupation was not mathematics; he pursued it as a hobby. During his leisure time, he made significant contributions to mathematics, with his most famous being the invention of logarithms. His work on logarithms began around 1594, and he carefully developed a system that enabled quick math calculations. Logarithms, in simple terms, measure the exponent you need to raise a specific number to get another number, for example if bx = y, then logby = x. It is a way of finding the unknown exponent in an exponential equation. 

In 1614, John Napier published his influential work on logarithms, titled “Mirifici logarithmorum canonis descriptio,” which translates to “A Description of the Wonderful Table of Logarithms.” The title reflects his goal of creating a “miraculous” mathematical tool. In this publication, Napier not only explained the mathematical intricacies but also introduced the term “logarithm” by combining the Greek words “logos” (proportion) and “arithmos” (number). This word became a permanent part of mathematics. Additionally, Napier differentiated between “natural” and “artificial” numbers and their corresponding logarithms, enhancing the clarity and depth of his ground-breaking work. 

Image shows a white background with two parallel black lines, labelled at intervals with numbers, letters, and greek letters
The two original parallel lines that John Napier used with moving particles.
Image credit: Mohsen Agheiboorkheili and John Giuna Kawagle, CC BY 4.0 DEED via Research Gate.

Napier’s idea of logarithms was based on a simple concept. He imagined two particles moving along two lines, one that went on forever and the other with a fixed length. Both particles started at the same point and began their journey simultaneously. On the never-ending line, the first particle moved at a constant speed, covering equal distances in equal time intervals. However, the second particle’s speed on the finite line depended on how far it had left to go to reach the end of that line segment. 

Image shows a white background with two parallel lines with dots labelled with letters. There are dotted lines connecting the dots on each line.
The relation between the two lines and the log and sine functions. The two particles and their line of travel can be clearly seen.
Image credit: Mohsen Agheiboorkheili and John Giuna Kawagle, CC BY 4.0 DEED via Research Gate.
Image is a picture of an old manuscript, brown aged paper background with a handwritten inked table filled in with numbers.
A page from Napier’s Mirifici logarithmorum tables, with trigonometric data for 34 degrees.
Image credit: public domain via Wikimedia Commons

Napier’s idea was like a clever game with two lines. Imagine one line that never ends and another with a fixed length. The distance left on the second line was like a “sine,” and the distance covered on the never-ending line was like the “logarithm” of that sine. As the sines got smaller, the logarithms got bigger. The way sines got smaller, and the way logarithms got bigger was different but followed specific rules.

Napier made a table that linked angles with special numbers. His table went from 0° to 45°, and it was symmetrical.

Napier undertook an incredible task, computing almost ten million entries, which he handpicked over approximately twenty years, starting around 1594. 

Logarithms, thanks to John Napier, have truly transformed our world. They play a vital role in diverse areas, from calculating compound interest and understanding exponential growth and decay, to determining the pH levels of substances and measuring the magnitude of earthquakes. Napier’s ingenious invention of logarithms has left an indelible mark on mathematics and science, making complex calculations more accessible and providing us with valuable tools for exploration and discovery. 

Image shows portrait of a man with a beard sitting in a wooden chair that is covered in red fabric. He is wearing a long black cloak with a tall white collar. The background is a dark brown with a crest in the top left corner.
John Napier
Image credit: public domain by Wikimedia Commons


Author
Gurminder Dhillon

Bibliography

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O’Connor, J.J. and Robertson, E.F. (1998). John Napier – Biography. [online] Maths History. Available at: https://mathshistory.st-andrews.ac.uk/Biographies/Napier/  [Accessed 25 Oct. 2023]. 

Wikipedia Contributors (2019). John Napier. [online] Wikipedia. Available at: https://en.wikipedia.org/wiki/John_Napier  [Accessed 25 Oct. 2023]. 

Unacademy. (n.d.). Logarithms and their Applications. [online] Available at: https://unacademy.com/content/nda/study-material/mathematics/logarithms-and-their-applications/#:~:text=Applications%20of%20Logarithms  [Accessed 25 Oct. 2023]. 

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