Napier’s bones were a practical tool for multiplication before calculators and slide rules. They were invented by John Napier and published in 1617 in *Rabdologiae*.

If you want the shortest answer: you choose the rods for each digit of the multiplicand, place them side by side, and use diagonal sums to build the product. For a source overview, see Napier’s bones on Wikipedia.

What Napier’s bones are

Napier’s bones are a set of numbered rods, usually one rod for each digit from 0 to 9. Each rod shows multiples of that digit arranged in a grid. The grid is split by diagonals so you can add digits along slanted lines instead of doing full multiplication in your head.

They were designed to make arithmetic faster and less error-prone. The method is especially useful when you need to multiply a number by another number with several digits.

A useful way to think about them is this: the rods turn multiplication into repeated addition with place value kept visible. That is why the tool was so handy before mechanical calculators became common.

The basic setup

To use the rods, you need the rods for the digits in your multiplicand. If you want to multiply 467 by another number, you place the 4, 6, and 7 rods in order next to each other.

Each rod contains rows for multiplying by 1 through 9. The tens digit is placed above the diagonal and the ones digit below it. If a product has only one digit, the tens part is zero. That layout is what makes the diagonal reading possible.

Here is the general idea:

  1. Pick the rods that match the digits of the top number.
  2. Place them side by side in the same order as the digits.
  3. Read the row for the bottom digit you are multiplying by.
  4. Add the numbers along each diagonal from right to left.
  5. Carry any extra tens to the next diagonal.

If you want a modern comparison, the process is a little like using a scientific calculator for a structured manual method: the steps are fixed, and you follow the layout instead of estimating.

A worked example: 467 × 8

Let’s multiply 467 by 8 with Napier’s bones.

First, place the rods for 4, 6, and 7 side by side. Then use the row for 8 on each rod:

  • 4 × 8 = 32
  • 6 × 8 = 48
  • 7 × 8 = 56

Written on the rods, these would be split by diagonals so the digits line up for addition. You then read the diagonals from right to left.

Read the diagonals

  • Rightmost diagonal: 6
  • Next diagonal: 5 + 8 = 13
  • Next diagonal: 2 + 4 = 6
  • Leftmost diagonal: 3

Now combine them from left to right, carrying when needed:

  • Leftmost digit: 3
  • Middle diagonal: 6
  • Next diagonal: 13, so write 3 and carry 1
  • Rightmost diagonal becomes 6 + 1 = 7

So the product is 3,736.

That is the basic skill with Napier’s bones: split each partial product into tens and ones, then add along diagonals with carrying.

Why the diagonals matter

The diagonal design is the whole trick. It keeps place value visible while you add several partial products at once. Without the diagonal split, you would have to remember where each digit belongs in the final answer.

This matters most when multiplying by a number with several digits. For example, if you multiply by 23, you use the row for 3, then shift one place and use the row for 2. The rods do not do the shifting for you automatically, but they make the structure easy to see.

That is also why Napier’s bones are often described as a bridge between manual multiplication and later calculating devices. They do not compute for you, but they organize the arithmetic. If you are interested in a later manual aid, the slide rule history shows how logarithms turned multiplication into another kind of shortcut.

How Napier’s bones fit into calculator history

Napier’s bones belong to a long line of tools built to reduce mental arithmetic. The timeline helps show where they fit:

  • The abacus was a counting frame used in many ancient cultures, with Roman and Chinese forms well documented.
  • John Napier published Napier’s bones in 1617.
  • William Oughtred’s circular and rectilinear slide rules date to about 1622, building on logarithms published by Napier in 1614.
  • Blaise Pascal built a mechanical adding machine from 1642.
  • Gottfried Wilhelm Leibniz presented a model of the stepped reckoner in 1673, aiming at multiplication and division by repeated shifted addition.
  • Charles Babbage proposed the Difference Engine in 1822 to tabulate polynomial functions by finite differences; the full machine was never completed in his lifetime.
  • Dorr E. Felt patented the Comptometer in 1887.
  • Bell Punch / Sumlock announced the ANITA Mk VII in 1961, generally described as the first all-electronic desktop calculator.
  • Hewlett-Packard released the HP-35 in 1972, the first handheld scientific calculator.
  • A 4-function pocket calculator became mass-market in the 1970s.

Napier’s bones sit early in this path. They are not a calculator in the modern sense, but they clearly point toward the same goal: reducing repetitive arithmetic. For a later step in that history, see Babbage’s Difference Engine.

How they compare with later devices

Napier’s bones are manual, while later machines automate more of the work.

  • Abacus: good for counting and basic operations, but it depends on the user’s skill with the frame.
  • Napier’s bones: help with multiplication by laying out partial products.
  • Pascaline: performs addition mechanically.
  • Stepped reckoner: designed to handle multiplication and division through repeated shifted addition.
  • Difference Engine: aimed at tabulating polynomial functions by finite differences.
  • Comptometer and ANITA Mk VII: keyboard-driven and electronic approaches that reduce manual setup.
  • HP-35 and pocket calculators: put complex arithmetic in your hand.

Napier’s bones are still useful as a teaching tool because they show what multiplication is doing under the surface. When you use them, you see the partial products instead of treating the answer as a black box.

A simple way to remember the method

The method is easier if you remember three words: place, read, carry.

  • Place the rods for the digits of the top number.
  • Read the row for the multiplier.
  • Carry when diagonal sums go past 9.

If the multiplication feels awkward at first, that is normal. The tool rewards careful setup more than speed. Once you understand one example, the same routine works for many others.

If you want to practice with place value in a different format, a Roman numerals converter can also help you think about how numbers are written and grouped.

Frequently asked questions

Who invented Napier’s bones?
John Napier published them in 1617 in *Rabdologiae*. They are named after him and were created as a practical aid for arithmetic, especially multiplication.
How do Napier’s bones work?
Each rod shows the multiples of one digit. You line up the rods for the number you want to multiply, then add along the diagonals for the chosen multiplier.
What is the main advantage of Napier’s bones?
They make multiplication more structured and reduce mental load. Instead of holding several partial products in your head, you read them from the rods and combine them step by step.
Are Napier’s bones the same as an abacus?
No. An abacus is a counting frame used for several operations, while Napier’s bones are rods designed mainly to simplify multiplication. They are related only in the broad sense that both help you compute manually.
What came after Napier’s bones?
The slide rule appeared about 1622, followed by mechanical calculators such as Pascal’s adding machine from 1642 and Leibniz’s stepped reckoner model from 1673. Later devices became more automatic.
Can Napier’s bones divide numbers?
They are mainly known for multiplication, though the same setup can support related arithmetic procedures. In practice, they are most useful when you want to see multiplication broken into smaller parts.
Why are they called “bones”?
The name comes from the original style of rods, which were often made from materials such as bone or similar hard material. The term stuck even when other materials were used later.

About this article

Written by Yahav Atia, WSO – the digital agency (wso.co.il) that owns and founded this site, with the help of AI, and passed automated quality checks before publication. Numbers in the worked examples were computed in code, not estimated.