## Introduction

In our modern digital age, it‘s easy to take for granted the incredible computing power we carry around in our pockets. With a few taps on a smartphone screen, we can perform mathematical feats that would have been unimaginable just a few generations ago. But the path to our current era of ubiquitous computation was a long and winding one, paved by visionary thinkers who dared to imagine a world where numbers could be tamed by machines.

One of the most pivotal of these figures was John Napier, a Scottish landowner and amateur mathematician. In 1617, Napier published a deceptively simple invention he called "numbering rods" – known today as Napier‘s bones. Little did he know, his elegant system for mechanical multiplication would lay the conceptual groundwork for the programmable computers that now power our world.

This article will dive deep into the story behind Napier‘s revolutionary invention. We‘ll explore how it worked, why it mattered, and how it helped shape the future of computation. Along the way, we‘ll see how a 17th century "computer" made of ivory and wood foreshadowed the electronic devices we use today, and how one man‘s passion for numbers changed the course of history.

## John Napier, Master of Numbers

To understand the significance of Napier‘s bones, we must first understand a bit about the man behind them. John Napier was born in 1550 to a wealthy family in the Scottish Borderlands. As a young nobleman, he received an elite education at the University of St Andrews, where he developed a keen interest in mathematics and theology.

Napier‘s greatest mathematical achievement was undoubtedly his work on logarithms, which he published in his 1614 book *Mirifici Logarithmorum Canonis Descriptio* ("A Description of the Admirable Table of Logarithms"). In it, he outlined a novel system for simplifying complex multiplication and division problems by converting them into additions and subtractions.

The crux of Napier‘s logarithms was a huge table of numbers that allowed him to essentially "look up" the solution to equations instead of computing them from scratch. As math historian Loren Graham explains in his book *The World of Mathematics*:

"Napier spent twenty years computing the enormously extensive tables … working through the night with his assistants and stopping only for prayers. The strenuous task proved too much for several of his associates; one died and another went insane."

This anecdote hints at the immense tedium and room for error inherent in manual mathematical calculations during the 17th century. It was to address this very problem that Napier began developing an even more ambitious project in the final years of his life: a physical tool to mechanize multiplication.

## Behold the Bones

Napier‘s solution was a set of rods, each engraved with numbers forming a unique multiplication table. When arranged together in a frame, the rods could be used to break down multiplication problems into simple addition by intelligently organizing the sub-products.

To multiply 8,749 by 6, for example, you would arrange the rods for 8000, 700, 40, and 9 in sequence, then locate the multiples of 6 on each rod. Adding up the resulting numbers along the diagonal (48, 0, 42, 54) would give the answer: 52,494.

*(Animation demonstrating how to multiply 8749 by 6 using Napier‘s bones. Via Wikimedia Commons)*

The ingenuity of Napier‘s system lay in how it transformed complex mental arithmetic into a straightforward physical procedure. A user could simply set up the initial conditions – the numbers to be multiplied – then "run the program" by following the fixed steps of rod placement and diagonal addition. The cognitive load of holding numbers in memory and calculating sub-products was offloaded onto the device itself.

As historian David Bryden puts it in his book *Napier‘s Bones: A History and Instruction Manual*:

"Napier‘s bones reduced the necessary mental effort of the user to a minimum. The user was not required to memorize tables or to understand any underlying mathematical concepts … The bones made multiplication a task that could be undertaken by anyone who could count."

This mechanical approach, where the "intelligence" resides in the system design rather than the user, would go on to be a core tenet of computer science. Napier‘s bones were, in essence, an algorithm in physical form – a repeatable process that reliably produced a desired output for a range of inputs. In this sense, they foreshadowed concepts like stored memory, sequential processing, and even parallel computing that would emerge in the centuries to follow.

## Crafting Calculation

The original sets of Napier‘s bones were hand-crafted from materials like ivory, brass, and boxwood. Creating them was a labor-intensive process that required great skill and precision to accurately transcribe the tables of numbers onto each rod.

According to historian Ivor Grattan-Guinness, a complete set of Napier‘s bones in the 17th century would have cost around 5-7 shillings – the equivalent of about £50-70 ($65-90) in today‘s money. At that price point, they were a luxury item affordable only to universities, government agencies, and wealthy individuals.

Over the next two centuries, however, the manufacture of Napier‘s bones became more streamlined and economical. By the mid-1700s, sets made of less expensive materials like wood, paper, and paste-board had brought the price down to a few pence, making them more widely accessible.

This reduction in cost and increase in availability mirrored later trends in the development of computing hardware. Just as mainframe computers gave way to minicomputers, then personal computers, then mobile devices, calculating tools followed a similar path of democratization as manufacturing improved and costs came down.

By the 19th century, Napier‘s bones had been eclipsed by more sophisticated calculating devices like the slide rule and mechanical calculator. But they remained popular as an educational tool for demonstrating core mathematical concepts. Even today, students around the world still learn to multiply using Napier‘s lattice method and make their own DIY Napier‘s bones sets out of paper or wood.

## The Legacy of Logs

While Napier‘s bones were undoubtedly a groundbreaking invention, it was his work on logarithms that would go on to have the most far-reaching impact on the field of computing. As Napier himself explained in the preface to *A Description of the Admirable Table of Logarithms*:

"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 realization was that multiplication could be mapped onto addition by creating a fixed table of logarithms – a lookup system where the sum of two logs equals the log of the original numbers‘ product. This insight paved the way for the development of slide rules, analog computers that could perform complex calculations by physically manipulating logarithmic scales.

More fundamentally, Napier‘s work hinted at the power of using one arithmetic operation (addition) as a proxy for another (multiplication). This concept of substitution would later become a cornerstone of digital computing, where all operations are reduced to sequences of binary addition.

Napier even anticipated the use of binary numbers themselves. In his 1617 treatise *Rabdologiae*, he outlined a system for representing all numbers using only the digits 0 and 1, writing that "it would be much more expeditious to use a dyadic arithmetic than a denary one." It would take nearly three more centuries for binary to be implemented in electronic computers, but Napier had planted the conceptual seed.

The impact of Napier‘s ideas can be directly traced through the work of later computing pioneers. Blaise Pascal, Charles Babbage, and John von Neumann all built upon Napier‘s foundations in developing their own calculating machines and frameworks for computation. As computer historian Doron Swade writes in his book *The Difference Engine: Charles Babbage and the Quest to Build the First Computer*:

"Napier‘s vision of mechanical computation, and the means by which he achieved that vision, provided a conceptual bedrock on which Babbage and his successors would build… His bones and rods, strips and cylinders… unlock a key transition in the prehistory of automatic computation."

## Conclusion

In our modern world of smartphones and supercomputers, it can be hard to imagine a time when even basic arithmetic was a mind-bending challenge. But Napier‘s bones offer a tangible reminder of the ingenuity and determined effort that went into developing the calculating tools we now take for granted.

Napier‘s invention was a testament to the power of outside-the-box thinking. By reimagining multiplication as a physical act instead of a mental one, he was able to devise a system that transcended the limits of human cognition. The bones were a bridge between the abstract world of numbers and the concrete world of objects, a way to make mathematical ideas manifest through human craft.

In this sense, Napier‘s bones embody the very essence of computing – the use of physical systems to represent and manipulate information. From the abacus to the iPhone, every computing device throughout history has relied on this fundamental principle of mapping meaning onto mechanism.

Napier‘s work also highlights the deeply human motivations behind even the most esoteric pursuits. He wasn‘t driven by dreams of fame or fortune, but by a pure love of numbers and a desire to expand the boundaries of what the human mind could achieve. As he wrote in *Rabdologiae*:

"I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Nearly four centuries after his passing, Napier‘s legacy endures in the cascading advancements his ideas helped set in motion. So the next time you reach for your phone to crunch some numbers, take a moment to remember the man who imagined a world where computation could fit in the palm of your hand. John Napier may not have lived to see the future he helped create, but his bones still speak to us across the ages – a testament to the timeless power of human ingenuity.