Napier's Bones — Definition, Formula & Examples
Napier's Bones is a manual calculating tool made of rods (or strips) that simplifies multiplication, division, and square roots by reducing them to simple addition. It was invented by Scottish mathematician John Napier and published in 1617.
Napier's Bones is a set of numbered rods, each inscribed with a column of the multiplication table for a single digit (0–9), arranged so that multi-digit multiplication is performed by reading and summing adjacent digits across the rods, effectively decomposing multiplication into a sequence of additions with carrying.
How It Works
Each rod displays the multiples of a single digit from 1 through 9, with each product split into a tens digit and a units digit separated by a diagonal line. To multiply a number, you place the rods for its digits side by side. You then read across the row corresponding to the single-digit multiplier, adding digits along diagonal strips from right to left and carrying when needed. The result of these additions gives you the final product.
Worked Example
Problem: Use Napier's Bones to multiply 46 × 7.
Step 1: Place the rods for digits 4 and 6 side by side. Look at row 7 on each rod. The rod for 4 shows 2/8 (meaning 28), and the rod for 6 shows 4/2 (meaning 42).
Step 2: Read the diagonal strips from right to left. The rightmost digit is the units digit of the 6-rod: 2. The middle diagonal combines the tens digit of the 6-rod (4) with the units digit of the 4-rod (8): 4 + 8 = 12. Write 2 and carry 1. The leftmost digit is the tens digit of the 4-rod (2) plus the carry (1): 2 + 1 = 3.
Step 3: Combine the digits from left to right to get the product.
Answer: 46 × 7 = 322
Why It Matters
Napier's Bones was one of the earliest mechanical aids for arithmetic and directly influenced the development of the slide rule and early mechanical calculators. Studying it helps you appreciate how place value and the distributive property underpin the multiplication algorithms you use today.
Common Mistakes
Mistake: Forgetting to carry when diagonal sums exceed 9.
Correction: After adding digits along each diagonal strip, check whether the sum is 10 or more. If so, write only the units digit and carry 1 to the next diagonal to the left, just like standard addition.