John Napier (1550-1617) was a Scottish landowner, mathematician, physicist, astronomer and theologian whose most important contribution was the discovery of logarithms. His logarithmic tables, published in 1614, revolutionized computation and helped accelerate the Scientific Revolution of the 16th and 17th centuries. Napier was also a pioneer in mechanical computation through inventions like Napier‘s bones. In many ways, he laid the conceptual foundation for modern computing.
Napier‘s Early Life and Education
John Napier was born into Scottish nobility on February 1, 1550 at Merchiston Castle near Edinburgh. The Napiers were a prominent family – John‘s grandfather Alexander Napier was killed fighting the English in 1547, and his father Sir Archibald Napier later became the 7th Laird of Merchiston.
Young John studied at home before enrolling at the University of St Andrews in 1563 at age 13. He studied Greek, Latin, logic and religion but left in 1567 without a degree, probably due to a friendship with a Catholic student during a time of Protestant Reformation in Scotland. Napier then studied in Europe for several years before returning to Scotland as a scholar in 1571.
Mathematics in the 16th Century
To appreciate Napier‘s innovations, it‘s important to understand the state of mathematics in his time. In the 16th century, calculations were still done using counting boards, Roman numerals and other cumbersome methods inherited from medieval times. There were no standard symbols for operations like plus, minus, multiply and divide. Fractions were written out in words.
Astronomical and navigational calculations in particular required working with very large numbers, often 6 or 7 digits long. Multiplying and dividing such large numbers was a laborious, time-consuming process. Mathematicians had begun to develop trigonometric tables to speed up calculations, but these too were limited by the difficulty of the underlying arithmetic. A simpler, more efficient method was sorely needed.
Napier‘s System of Logarithms
This was the problem Napier set out to solve with his system of logarithms, which he developed through decades of experimentation and described in his 1614 book Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms).
Napier defined logarithms as a way to convert multiplication problems into easier addition problems, and division into subtraction, by relating arithmetic sequences to geometric sequences. As he wrote in the preface to the Descriptio:
Seeing there is nothing that is so troublesome to mathematical practice…than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.
Napier‘s key insight was that if a sequence of numbers increased arithmetically (by a constant amount) while another sequence increased geometrically (by a constant multiple), the two sequences could be mapped to each other, allowing multiplication in the geometric sequence to be achieved by addition in the arithmetic sequence.
For example, consider these two sequences:
| Arithmetic | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Geometric | 1 | 2 | 4 | 8 | 16 | 32 |
The arithmetic sequence starts at 0 and increases by 1 at each step, while the geometric sequence starts at 1 and doubles (multiplies by 2) at each step. We can map each number in the arithmetic sequence to the corresponding number in the geometric sequence. Then to multiply numbers in the geometric sequence, we can:
- Find the corresponding numbers in the arithmetic sequence
- Add those arithmetic numbers together
- Find the result in the arithmetic column
- Return the mapped geometric number
For instance, to multiply 4 and 8, we first find the arithmetic numbers mapped to them (2 and 3). We add these to get 5, then find the number mapped to 5 in the geometric column, which is 32. So 4 x 8 = 32.
Napier called the numbers in the arithmetic sequence "logarithms" and published tables mapping them to the geometric sequence. But rather than the neat doubling sequence above, he used a geometric sequence starting with 10,000,000 and decreasing by a factor of (1 – 1/10,000,000) each time, while the arithmetic sequence decreased from 0 to large negative numbers.
Here is a small sample of Napier‘s original logarithm table:
| Number | Logarithm |
|---|---|
| 10000000 | 0.000 |
| 9999999 | 0.000000100 |
| 9999998 | 0.000000200 |
| … | … |
| 9995001 | 0.049950012 |
| … | … |
| 9900473 | 0.995002498 |
So to multiply 9,995,001 and 9,900,473 using Napier‘s tables, we would:
- Find the logarithms of those numbers:
- 9,995,001 maps to 0.049950012
- 9,900,473 maps to 0.995002498
- Add the logarithms: 0.049950012 + 0.995002498 = 1.044952510
- Look up the result in the Number column: the closest entry is 1.044952471, which maps to 9,895,525.
As we can see, there are some approximations involved, and the calculations are still a bit cumbersome by modern standards. But Napier‘s tables reduced weeks of effort to hours and were a major breakthrough, quickly spreading throughout Europe.
Refining Logarithms
In 1615, the English mathematician Henry Briggs traveled to Edinburgh to meet with Napier and discuss his system. Together, they refined the logarithms to use base 10 instead of Napier‘s unwieldy decreasing geometric sequence.
Briggs published improved base-10 logarithm tables in 1624, after Napier‘s death. These "common logarithms" became the standard in science and mathematics for centuries until the advent of the digital calculator in the 1970s.
Common logarithms further simplified calculations because log10(10n) = n. This means numbers could be expressed in scientific notation as a coefficient times a power of 10, and the logarithm of that number would simply be the exponent. No table lookup needed.
The unit of measurement named after Napier, the Neper (Np), is based on natural logarithms and is used in acoustics, electronics and other fields dealing with ratios and levels. The decibel (dB) is one-tenth of a Neper:
1 Np = ln(e1) ≈ 8.685889638 dB
Napier‘s Bones
In addition to his work on logarithms, John Napier also developed mechanical aids to calculation, most famously Napier‘s bones. This was an abacus-like device consisting of a base board with a rim, and movable rods engraved with multiplication tables.
To use Napier‘s bones, a number was entered on the rods and the corresponding multiples read off the rods and added together. This sped up tedious multiplications and divisions.
For example, to multiply 48,642 by 5 using Napier‘s bones, we would:
- Enter 48,642 on the rods
- Read off the multiples from the "5" row:
- 5 x 4 = 20
- 5 x 8 = 40
- 5 x 6 = 30
- 5 x 4 = 20
- 5 x 2 = 10
- Add diagonally, carrying any tens to the next diagonal: 243,210
Napier‘s bones and other mechanical calculators like the slide rule (which also used logarithmic scales) were the cutting edge of computing technology well into the 20th century, until they were made obsolete by electronic computers.
Napier the Digital Innovator?
Although John Napier lived some 400 years ago, in many ways his work foreshadowed the development of modern digital computing. His system of logarithms boiled down multiplication and division of large numbers to simpler addition and subtraction. This is similar to how computers perform complex calculations through repeated simple operations.
Napier‘s bones and other calculating machines that followed were early examples of automating mathematical functions in hardware. The binary arithmetic that powers today‘s computers also relies on converting between number systems, similar to how logarithms map numbers to and from a different representation to simplify calculations.
One wonders what Napier would have thought of the supercomputers and smartphones of today, or how his brilliant mathematical mind might have pushed computer science forward had he been born a few centuries later. While we can only speculate, it‘s clear that Napier‘s innovations in computing laid important groundwork for the Information Age.
Legacy and Impact
John Napier died on April 4, 1617 at age 67 and was buried at St Cuthbert‘s Church in Edinburgh. In his lifetime, he believed his theological writings against the Catholic church would be his great legacy. But it was Napier‘s mathematical work, particularly the development of logarithms, that secured his lasting fame and impact.
Napier‘s logarithms revolutionized the tedious process of mathematical calculation and were quickly adopted by scientists and navigators across Europe and beyond. As the English mathematician William Oughtred wrote in the preface to his 1647 book Clavis Mathematicae (The Key of the Mathematics):
The admirable table of logarithms, first published by the never sufficiently praised John Napier baron of Merchiston… By the aid of these, multiplication of numbers is reduced to addition, division to subtraction, extraction of roots to easy divisors, raising of powers to easy multiplications. Without these last four operations, the more abstruse parts of mathematics can make no progress…
Napier‘s discovery also spurred further developments by other mathematicians like Henry Briggs, Johannes Kepler and Leonhard Euler. Logarithms became a foundational tool of mathematics and science through the 20th century. Even in today‘s age of digital computers, logarithms have important applications in computer science, physics, chemistry, biology and many other fields.
More broadly, Napier‘s work as an innovator exemplified the Scientific Revolution taking place in Europe in his era. Like other pioneers of the time, Napier used experimentation, analysis and mechanical invention to advance human knowledge. Rather than deferring to ancient authorities, he devised practical solutions to the mathematical challenges of his day.
John Napier‘s legacy as a mathematician and computational pioneer remains strong. He has lent his name to Edinburgh‘s Napier University, the Napier Professor of Mathematics chair at the University of Edinburgh, the Napier crater on the Moon, and even a brand of Scotch whisky. But his most important memorial is the ongoing advancement of science and technology that his discoveries helped make possible.
