Logarithms: The Long Forgotten Story Of Scientific Progress

Computing logarithms was not my most favourite activity at school. In fact, I disliked logarithms because of their relative lack of clarity and transparency. I knew what I was doing. But I barely understood why I was doing it. Questions like, “Why do I have to invoke logs and lookup annoying tables to solve simple problems?” and “How did the person who originally derived this equation know when and where to invoke the logarithm?” kept popping up in my head.

As time went on, access to calculators and computers steadily increased. As a result, my curiosity went on the back burner. It was much later in my scientific journey that I started understanding the real genius behind logarithms. Shortly after, I found myself delving into the history behind the invention.

“If only I was taught this at school…”, I thought to myself. The story of the logarithm is so good that I simply had to share it. That is exactly what brings you to this article. So, sit back, relax, and enjoy the pleasant ride!

This essay is supported by Generatebg

Challenges of the late Sixteenth Century

There were many challenges that people faced during the late sixteenth century. Rampant conflicts and political instabilities aside, we shall narrow down our focus on the challenges faced by the mathematicians back then. Astronomy was quite popular during this time, and the astronomers were facing difficulties dealing with, well, ‘astronomical’ numbers.

Multiplying and dividing large numbers took a very long time for the astronomers of the time (even months). So did finding squares, square roots, cubed roots, etc., of the astronomical numbers. Furthermore, the methods used back then were so tedious that they were error-prone. So, quite a few mathematicians among the astronomers set out to solve this problem. That’s where our first protagonist comes in.

---

The Scottish Polymath

John Napier of Merchiston was born in 1550 into royalty. He seldom had a shortage of resources and was known for his intellect. Not much is known about his education, except that he left Scotland for mainland Europe during his youth for education. When he returned, he spoke fluent Greek. Greek was not taught in many European universities back then.

Being an astronomer himself (among others), Napier set out to solve the challenges with astronomical numbers. He wanted to solve them faster, with more precision, and with fewer errors. He invented a device called Napier’s Bones to calculate the products and quotients of numbers. This was a handheld calculator of sorts.

Being a perfectionist at heart, even this invention did not satisfy him. He was constantly on the lookout for newer and better ways of calculating large numbers.

The Swiss Clockmaker

Swiss watches would not have their reputation today if it were not for genius clockmakers such as Jost Bürgi. Born in 1552 in Toggenburg, not much is known about his education either. However, he turned out to be a genius clockmaker, a maker of astronomical instruments, and a mathematician. He served at the courts in Kassel and Prague.

Bürgi was celebrated for his mechanical astronomical models and his invention of a table of sines (Canon Sinuum). Such tables were critical for sea navigation back then. He had a keen eye for numbers and calculations.

Although he was a genius at his craft, Bürgi largely kept his work to himself and seldom shared his knowledge. This, however, changed when he started working closely with the famous astronomer, Johannes Kepler. If it wasn’t for Kepler, we might not have come to know about Bürgi’s brilliant contribution to logarithms.

---

Logarithms — History in the Making

Back in Scotland, Napier was still after better and faster methods of calculating numbers. He was fascinated by Prosthaphaeresis, a method used to quickly approximate multiplication and division using trigonometry. He was generally convinced by the idea of using abstractions to reduce computation.

He noticed that when a sequence of powers of a number are multiplied, their powers (now known as exponents) form an arithmetic sequence.

As fundamental as it may seem now, this insight formed the basis of the concept of logarithms. 

Napier published his book Mirifici Logarithmorum Canonis Descriptio in 1614. This book contained fifty-seven pages of explanation and ninety pages of natural logarithm tables. So, the term ‘Logarithm’ was coined by Napier.

In the meantime, Bürgi had invented a method of speeding up calculations of his own. The idea itself was comparable to Napier’s, but his approach was different. Unlike Napier, Bürgi kept his set of tables to himself and used them to do his own calculations. But upon persistent requests from Kepler, Bürgi published his work, Progress Tabulen, in 1620, 6 years after Napier.

This book revealed that Bürgi clearly invented his tables independently of Napier’s work. In fact, Bürgi’s tables are now understood as antilogarithms. You could say that both Napier and Bürgi approached the same problem from different angles. Further development of the concept of logarithms from this point, however, took the path of Napier’s and not Bürgi’s.

---

The Principle of Logarithms

A logarithm essentially asks the following question: To what power must a number be raised to get a particular result.

In the following example, the logarithm operation performed on 64 with base 2 results in 6. In other words, it asks 2^x=64, where x is the result of the logarithm operation.

The idea is that we precompute powers of 2 as logarithms. Then, we compute complex multiplications (such as 2²⁵ * 2³⁰ = 33554432 * 1073741824) as additions, and then work out the product from the log tables.

This is what we would call ‘reducing the order of an operation’ in today’s terms. But hang on a minute! What about numbers that are not powers of 2? How do we multiply those? That’s where our last protagonist comes in!

The English Mathematician

Henry Briggs was an English mathematician who was known for his academic achievements. He came across Napier’s work when he was working as a lecturer at Gresham College, London. He was impressed with Napier’s work, and this triggered his imagination.

It is said that he lectured his students on the notion of the choice of base for logarithms. Soon after realizing the potential of this insight, he wrote Napier, and the two agreed to meet. In 1615, they met and Briggs suggested that Napier make the logarithm of 10 equal to 1.

He essentially suggested that Napier switch to base 10 logarithms. This was a genius suggestion because of the following two reasons:

1. One needed to compute logarithm values for numbers only up to 10.

2. Beyond 10, any number can be expressed as a multiple of 10.

It is perhaps easier to look at an example of this:

Napier saw the merit of this change and agreed to it. Just two years later, in 1617, Napier passed away due to illness. After this, Briggs took it upon himself and continued to develop the concept of logarithms. Eventually, Jacob Bernoulli discovered the natural constant ‘e’, and this thrust an entirely new dimension of applications for logarithms.

---

Logarithms Today

Fast forward to today, it is easy to think that the development of computational technology would have made logarithms obsolete. But this is simply not the case. Brilliant ideas do not die easily!

The logarithm of today has been perfected by many mathematicians over the years and has become its own function. It is now known as the inverse of the exponential function. Any real-life phenomenon that involves exponential decay or growth involves logarithmic functions. For example, studies involving the rise and decline of populations or the spread of pandemics involve logarithms.

Logarithms also represent human intuition as to how we view change. Phenomena that exhibit exponential changes do not make intuitive sense to us. Applying logarithmic scales to such phenomena helps us linearise their perception (humans are natural linear thinkers). Examples of such scales include the Decibel (used to measure the intensity of sound) and the Richter scale (used to measure the intensity of earthquakes).

While we think linearly, we perceive the world logarithmically. We are complex creatures, after all. This phenomenon is explained by the Weber-Fechner law, and logarithms are at the centre of understanding it. If this phenomenon interests you, I have written an independent article explaining it. But for now, think of the following examples:

1. The older we get, the faster each year seems to go by.

2. The richer we get, the lesser money seems to satisfy us.

Both of these behavioural patterns can be modelled using logarithms.

What Next?

Born from such rich history, I find it fitting that people who use logarithms also get to learn the history behind them. For us users, a logarithm is merely a tool. For most students, logarithms are ‘one of those’ boring functions. But for the geniuses of the past, it was (arguably) their entire life’s work. They were trying to solve a painful problem for humanity.

“..(Napier’s logarithms).. by shortening the labours (have) doubled the life of the astronomer.” 

— Pierre Simon Laplace.

We walk on the shoulders of giants! It is time for us to respect this fact and remain thankful for their contributions.

---

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

I hope you found this article interesting and useful. If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you: Why Earning More Leads To Lesser Satisfaction,  The Thrilling Story Of Calculus and Why Do You See Mirrors Flipping Words?