Napier's Bones Calculator

Napier’s bones do no arithmetic. Lay the rods for 6785 on the board, look along the eighth row, and nothing has been multiplied: eight little products were engraved there long ago and are simply lying next to each other. What the device does is spare you the times table. The addition, and the carrying, it hands straight back.

That sounds like faint praise, and one of its best historians meant it as worse. Writing in the 1911 Britannica, the mathematician J. W. L. Glaisher judged that the rods would be of little use to anyone who knew the table up to nine times nine, and that the delight they caused shows how rough everyday arithmetic was in 1617. He was right about the arithmetic. He was less fair to the idea, which is cleverer than it looks and worth following step by step.

A book for people who would rather not use logarithms

John Napier was born in 1550 at Merchiston Castle in Edinburgh and died in that city on 4 April 1617. Three years before his death he had published the logarithms that made his name. In his last year he printed a small Latin book, Rabdologiae, at Edinburgh, dedicated to Alexander Seton, Earl of Dunfermline, the chancellor of Scotland. Glaisher describes it as a duodecimo of one hundred and fifty-four pages.

The book holds three inventions. The first is the set of numbering rods. The second is a box of metal plates Napier called the promptuary, for long multiplications. The third is an arithmetic done with counters on a chessboard, which rests on writing numbers in the scale of two. Napier says he devised all three for those who would prefer to work with ordinary numbers and leave logarithms alone.

His dedication gives the reason for printing. The rods had pleased the friends he showed them to, were already close to common use and had been carried abroad, and he was advised to publish before they appeared under another name. In the book they are virgulae, little rods. A manuscript of his calls them bones, and MacTutor’s account says the sets were made of ivory, which is how the nickname stuck. William Leybourn’s English handbook of 1667 put both names in its title.

What is on a rod

A rod has a digit at its head and nine squares below. Each square holds the next multiple of that digit, and each is cut by a diagonal from its bottom left corner to its top right. The tens go above the line and the units below. On the seven rod the fourth square reads 2 over 8. A multiple under ten leaves its upper half empty.

An eleventh strip, the index, numbers the rows from one to nine. Set rods beside it and row six shows six times each digit at once: the right entries of the times table, side by side.

Napier wanted the set to be compact, so his rods are square in section and written on all four sides. Wikipedia’s article puts it as 40 tables on 10 rods. Glaisher lists the faces: the first rod carries the multiples of 0, 1, 9 and 8, the second of 0, 2, 9 and 7, and so on to the tenth with 3, 4, 6 and 5. Two faces of every rod carry the complements to nine of the other two, written the other way up. With one set, he notes, any number below 11,111 can be laid out.

Adding along the slant

Now the reading. Take the worked case from the Wikipedia article: 6785 x 8 = 54280. Rods 6, 7, 8 and 5 lie beside the index, and row eight shows 4 over 8, 5 over 6, 6 over 4 and 4 over 0.

Start at the right. The last triangle holds 0, so write 0. The next lane runs across the join between two rods: the 4 below the diagonal on the eight rod and the 4 above it on the five rod make 8. The lane after that is 6 and 6, which is 12; write 2 and carry 1. Then 8 and 5 and the carried 1 make 14; write 4 and carry 1 again. The first triangle holds 4, and with the carry it becomes 5. Read back the digits: 54280.

Each lane holds two digits at most, one from each side of a join, so a lane can never pass 18 and the carry is never more than 1. That small fact is the whole comfort of the device. It is also why the diagonals are there at all: they pair every units digit with the tens digit that belongs to the same place.

A multiplier with several digits needs a row for each. For 825 times 913 the article reads row three as 2475, row one as 825 and row nine as 7425, writes each one place further left than the last, and adds the column by hand: 753225. The rods are no help with that final sum.

Division reuses the same rows. Set the rods to the divisor and the nine rows are its first nine multiples. Take the largest that fits under the leading digits of the dividend, write its row number as a digit of the quotient, subtract, bring down the next digit, and repeat. The article divides 46785399 by 96431 this way and gets 485 with 16364 left over.

Older than the rods, and stranger at the edges

None of this was new in 1617 except the cutting into strips. David Eugene Smith’s history says plainly that the rods are arranged to represent the gelosia method of multiplication, named for the window grating its diagram resembles. In that method the two numbers stand along the top and side of a grid, every cell is split by a diagonal and filled with a small product, and the diagonals are summed.

Where the grid came from is told differently by different writers. Smith thought it was very likely developed first in India, travelled to China and into Arab and Persian works, and passed from the Arabs to Italy, where it fills manuscripts of the fourteenth and fifteenth centuries and first reaches print in the Treviso arithmetic of 1478. The Wikipedia article on the method, citing a later history of algorithms, gives the earliest recorded uses as an Arab text of the late thirteenth century, a Latin treatise written in England around 1300, and a Chinese work finished in 1450. It adds that the credit often given to al-Khwarizmi or to Fibonacci is mistaken.

Napier pushed his rods further than the grid had gone. For square roots he added a plate with 3 columns: the squares from 01 to 81 in cells cut like the rods, the even numbers from 2 to 18, and the digits 1 to 9. The other face of the plate serves cube roots.

The method runs like long division. Mark the number off in pairs. The largest square under the first pair gives the first digit of the root. Its even number goes on the board as rods, and from then on each row, read across those rods and the squares column, is a candidate to subtract. For 46785399 the rows chosen are 36, 1024, 4089 and 123021, the root is 6839, and 13478 is left over.

Makers kept reshaping the thing. The Science Museum Group holds a set of about 1690 in a boxwood case, 57 by 95 by 20 millimeters overall, a French set of eleven card strips engraved in 1673, and a box of ten figured rollers. Its catalogue notes that the roller form was first published by Gaspard Schott in 1668, so an inscription inside the lid claiming the box was Napier’s own is unlikely to be true. Smith records the rods reaching China by the early eighteenth century and Japan in the century after.

Why the carry was the real problem

Every later improvement attacked the same weakness. Denis Roegel’s survey lists rods slanted so that the two digits of a lane stand one above the other, a colored set of 1738, and Schickard’s calculating machine of 1623, which used the rods as an aid. In none of them were the carries of a row done for you.

The fix came from a French railway engineer, Henri Genaille. Roegel traces his method to a meeting in 1878, has Edouard Lucas describing the rods in 1884, and dates the printed sets to 1885. Other accounts, the Wikipedia article among them, give 1891. On these rulers each cell lists every units digit it could show, one for each possible carry coming in, and a printed triangle leads from the digit you are on to the right digit on the next rod. You follow the triangles from base to tip and read the product. No addition is left.

By then it hardly mattered, because Napier had beaten his own rods before he printed them. Logarithms, published in 1614, turn a whole multiplication into one addition and a division into one subtraction, whatever the length of the numbers. Henry Briggs printed his first work on them in 1617 and the logarithms of thirty thousand numbers to fourteen places in 1624. Edmund Gunter put a logarithmic scale on a two foot ruler in 1620, and by 1632 William Oughtred was using two such rulers together, with no dividers needed. Smith’s verdict is that the rods might have mattered for trigonometric work if logarithms had not existed.

So be clear about what the bones cannot do. They give the multiples of one number, and only as pairs of digits still to be added. They do not carry, they do not add the partial lines, and they say nothing about where a decimal point goes. A slip in one lane is silent. What they offer in return is exactness: every digit of 54280 is right, where a table of logarithms stops at its last printed place.

That is also why they are still worth an afternoon. Cut a set from card, lay out a number, and the place value system is suddenly visible as geometry: units leaning toward the tens beside them, each join a column, each carry a single step to the left. Napier wanted to spare people the labour of calculating. The rods spare less of it than he hoped, and show more of it than almost anything since.