What is the calculating device for bones? The story of Napier's Rods

Invented by Scottish mathematician John Napier in the early 17th century, a manual calculating tool simplified complex arithmetic for a generation of scholars and merchants. This ingenious device, famously known as the calculating device for bones or Napier's Rods, was a significant step toward mechanized computation.

Quick Summary

Napier's Bones is a 17th-century manual calculating tool consisting of rods or strips inscribed with multiplication tables, used to perform complex multiplication and division through a process of lattice multiplication.

Key Points

• Inventor: Scottish mathematician John Napier invented the device in the early 17th century to simplify calculations.

• Function: The rods are used for manual multiplication, division, and finding square roots, using a method called lattice multiplication.

• Materials: The name "Napier's Bones" comes from the materials often used to create the rods, such as bone or ivory.

• Mechanics: The device consists of a set of rods or strips, each featuring a multiplication table for a single digit.

• Legacy: This manual tool was a forerunner to more complex mechanical calculators and influenced the development of the slide rule.

The Origins of Napier's Bones

Before the advent of modern electronics, computation was a laborious and error-prone process. John Napier, a nobleman and mathematician from Scotland, sought a way to ease these difficulties. In 1614, he published Mirifici Logarithmorum Canonis Descriptio, which introduced logarithms, a method to turn multiplication into addition. Later, in his 1617 work Rabdologiae (from the Greek for "rod-reckoning"), he presented a practical, physical manifestation of his ideas: a set of numbered rods, which came to be known as Napier's Bones.

Napier created the device specifically to mechanize and speed up multiplication and division. The use of materials like bone, ivory, or wood for the rods is where the name derives, making it a fitting answer to the question, "What is the calculating device for bones?" This innovation was rapidly adopted across Europe, making tedious calculations more accessible to a wider audience.

How the Device Works: A Step-by-Step Guide

Napier's Bones are based on a system of lattice multiplication. Each bone or rod contains a multiplication table for a single digit, from 0 to 9. The digits of the product are written in a small square, with a diagonal line separating the tens digit (above) from the units digit (below).

To multiply a multi-digit number by a single-digit number, you follow these steps:

1. Arrange the Rods: Place the rods corresponding to the digits of the multi-digit number side-by-side in a rack or frame. For example, to multiply 425, you would line up the rod for 4, then 2, and then 5.
2. Use the Index Rod: Locate the row on the index rod (a special rod with the digits 1-9) that corresponds to the single-digit multiplier.
3. Read the Results: Read the digits from the corresponding row on the arranged rods. The sum of the digits in each diagonal section gives the digits of the product.
4. Perform Addition: Starting from the rightmost diagonal, add the numbers. If a sum is greater than 9, carry the tens digit over to the next diagonal column, just as in traditional addition.

Example: Multiplying with Napier's Bones (425 x 6)

To illustrate, let's multiply 425 by 6:

• Setup: Arrange the rods for digits 4, 2, and 5 next to each other.
• Multiplication: Read the numbers from the 6th row of the arranged rods.
  • Beneath the 4, the 6th row shows 2/4 (24).
  • Beneath the 2, the 6th row shows 1/2 (12).
  • Beneath the 5, the 6th row shows 3/0 (30).
• Addition (from right to left):
  • The first diagonal has 0. Result digit is 0.
  • The second diagonal has 3 and 2. Sum is 5. Result digit is 5.
  • The third diagonal has 1 and 4. Sum is 5. Result digit is 5.
  • The fourth diagonal has 2. Result digit is 2.
• Final Result: Reading the digits from left to right, the result is 2550.

The Legacy of Napier's Bones

While a significant advancement, Napier's Bones were a transitional technology. They simplified the hard parts of multiplication and division but still required manual addition to get the final answer. However, their impact was profound, proving that complex arithmetic could be broken down into a series of more manageable, repeated steps. This principle of automated or semi-automated calculation paved the way for more complex devices.

The invention's influence can be seen in later instruments such as the slide rule, which was developed around the same time and used Napier's discovery of logarithms. It also inspired the creation of the first true mechanical calculators. Wilhelm Schickard, for example, incorporated a version of Napier's Bones into his "calculating clock" in 1623, one of the earliest machines to automate all four basic arithmetic operations.

Comparison: Napier's Bones vs. The Abacus

Feature	Napier's Bones	Abacus
Mechanism	A set of inscribed rods based on lattice multiplication.	A frame with beads that slide on rods or wires.
Core Operations	Primarily multiplication and division.	All four basic arithmetic operations (addition, subtraction, multiplication, and division).
Complexity	Reduces multi-digit multiplication to a series of single-digit additions.	Requires a learned system of bead manipulation for calculations.
Origin	Early 17th century Europe.	Ancient origins, with roots in Mesopotamia and widespread use across various cultures.
Material	Typically wood, bone, or ivory.	Varies widely, from wood frames and clay beads to metal and plastic.

Advanced Operations with Napier's Rods

Napier also outlined methods for performing more advanced calculations with his bones, including division and the extraction of square roots. For division, the process is essentially the inverse of multiplication, involving a series of subtractions and trial-and-error using the rods to find the correct quotient. Though more complex, it was still a considerable time-saver compared to purely mental calculation.

Conclusion: A Stepping Stone to Modern Computing

The story of Napier's Bones is a testament to human ingenuity in solving mathematical problems. This simple yet effective device, made from materials as basic as bone, represents a crucial stepping stone in the history of calculation. It provided a glimpse into a future where computation could be mechanized, influencing a long line of inventors who would eventually build the foundations of the digital age. From the 17th-century merchant to the modern software engineer, the quest for faster and more reliable calculation methods has been a constant in human history, and Napier's Bones hold an important place in that narrative.

For more detailed historical context on this pivotal invention, you can visit the Wikipedia page on Napier's Bones.

Frequently Asked Questions

What is the calculating device for bones, and who invented it?

The calculating device for bones is Napier's Bones, invented by Scottish mathematician John Napier in the early 17th century to simplify complex arithmetic like multiplication and division.

Why is the calculating device for bones called 'bones'?

The device was named 'bones' because the individual rods were often crafted from materials like bone or ivory. The term 'bones' is a literal reference to the physical material used in their construction.

How did Napier's Bones work for multiplication?

The device worked by arranging rods corresponding to the digits of the number to be multiplied. By reading the numbers from the appropriate row and adding the digits along diagonal sections, a user could quickly find the product, based on the principle of lattice multiplication.

Was Napier's Bones a fully automatic calculator?

No, it was not fully automatic. While it significantly simplified multiplication and division, users still had to perform manual addition on the diagonal sections to arrive at the final answer. It was a manual computing aid, not a mechanical calculator.

How did Napier's Bones influence later inventions?

Napier's Bones played a crucial role in the history of computing. It influenced the design of early mechanical calculators, like Wilhelm Schickard's 'calculating clock,' and its underlying mathematical principles contributed to the development of the slide rule.

Can Napier's Bones be used for other calculations besides multiplication?

Yes, John Napier also described methods for using the rods to perform division and even to extract square roots, although these processes were more complex than simple multiplication.

Is Napier's Bones related to modern bone density scans?

No, Napier's Bones is a historical mathematical instrument and has no relation to modern medical devices like bone density scanners (DEXA scans). The similarity in wording is a simple historical coincidence.