Napier’s Bones

1. Napier’s Bones An Adventure in 17th Century Arithmetic by Mr. Hansen

2. John Napier John Napier, a 16th Century Scottish scholar,contributed a host of mathematical discoveries, among them the world’s first calculator. John Napier (1550 – 1617)

3. He is credited with creating the first computing machine, logarithms and was the first to describe the systematic use of the decimal point. Other contributions include a mnemonic for formulas used in solving spherical triangles and two formulas known as Napier's analogies. “In computing tables, these large numbers may again be made still larger by placing a period after the number and adding ciphers. ... In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many ciphers after it as there are figures after the period.”

4. High tech in the 17th century, was what we’d now call basic astronomical arithmetic calculations, all done by hand. It took Johannes Kepler (1571-1630) nearly 1000 large pages of dense arithmetic do discover the laws of planetary motion! A typical page from one of Kepler’s notebooks Johannes Kepler (1571-1630)

5. Napier’s Bones In 1617, the last year of his life, Napier invented a tool called “Napier's Bones” which reduces the effort it takes to multiply numbers. This was a time when few people could multiply beyond 5 (x) 5. “Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”

6. Napier’s bones were called that because they were often made of bone, ivory, silver, or wood. The were were universally popular and common until the late 1800s. Sometimes the Napier tables were engraved on rods in a case so that numbers could be “dialed in”.

7. Napier’s bones make multiplication and division easier. Multiplication and division are reduced to simple addition, although a pencil and paper are required. This boxed set has ten rods, allowing computations up to 100,000,000. The left (or “index”)rod is fixed to the case. It is numbered from 1 to 9. The movable rods are numbered at the top. The numbers down them rods show the product of the number at the top times the corresponding numbers on the index rod. Here the “3” rod shows three times each of the numbers on the index rod.

8. The bones are easy to use. Multiplication and division are set up the same way. Set the problem up by laying down rods corresponding to the number being multiplied or divided. This setup shows the number 3579 which we will show being multiplied by 43.

9. This is the problem shown on our “paper bones.” The “3”, “5”, “7” and “9” strips are set up next to the index.

10. Using a pad of paper, we write down the individual products of 40 and 3 times each digit of 3579. The results are: Multiply by 40: 4 Times : 3 5 7 9 is 12 20 28 36 Adjust carries 1 4 3 1 6 --------------------------------------------------------- Shift to the right one decimal place and multiply by 3: 3 Times 3 5 7 9 is 9 15 21 27 Adjust carries 1 0 7 3 7 --------------------------------------------------------- Add results 1 5 3 8 9 7 (Adjust carries if necessary)

11. Divide 75159 by 3579: Set the rods of the divisor 3579, giving tables of 1 (x) 3579 to 9 (x) 3579 or 10 (x) 3579 to 90 (x) 3579. First digit: 10 (x) 3579 to 90 (x) 3579 Row 2: 20 (x) 71580 Row 3: 30 (x) 107370 meaning the dividend is between 20 and 30. The first digit is 2. Subtract: 75159 (-) 71580 3579 <- The remainder is now the dividend. Second Digit: The table now used as 1 (x) 3579 to 9 (x) 3579 Row 1: 3579, so the second digit is 1, and the solution is 21.