Shady Char­ac­ters advent calendar 2023: Napier’s bones

Welcome to day three of the first ever Shady Characters advent calendar! I’m counting down to Christmas by way of a collection of beautiful, clever, important, and/or outright odd calculators and calculating devices. Some come from the pages of Empire of the Sum, some are part of my Calculator of the Day series, and some will be new to the blog. I won’t manage twenty-four posts, but I do plan to hit at least one every other day. I hope you enjoy the series!

The life of John Napier of Merchiston, a Scottish laird who lived from 1550 to 1617, is notable for many reasons. He was a fanatical Protestant for one thing, having written an infamous anti-Catholic screed entitled The Plaine Discovery of the Whole Revelation of St. John. He was also an alchemist, dabbling in the manufacture of the so-called philosopher’s stone, a mythic substance capable of all kinds of miracles. (He failed to make any, but then he was hardly the only one.)1

But Napier is best know for his invention of logarithms: an appropriately esoteric mathematical construct that allows difficult multiplications to be decomposed into simple additions. With a book of logarithms in hand, an astronomer (or, as like, an astrologer) could grapple with the motion of the spheres far more easily than before. It was a discovery that rippled across the mathematical firmament and made Napier one of its stars.2

Logarithms would soon be used by others to invent a multiplying device — the simple, ingenious slide rule — that would become indelibly associated with maths and engineering. But although Napier’s name is often spoken in connection with the slide rule, there is another device that he himself created, and which bears his name directly: Napier’s bones.

A set of Napier's bones, or rods, likely manufactured in London in the early twentieth century.
A set of Napier’s bones, or rods, likely manufactured in London in the early twentieth century. (CC BY 4.0 image courtesy of Nisse Cronestrand at the Tekniska Museet.)

So called because some examples were made from bone-white ivory, Napier’s bones were a set of wooden rods inscribed with numbers along their length. To use them, one arranged the rods to match the multiplicand, or first number to be multiplied, and then read off individual results along the length of the rods according to the multiplier, or second number to be multiplied. The individual results were then summed, taking care to carry tens where necessary.3 And as fiddly as this sounds, Napier’s bones did make large multiplications considerably easier to carry out.

Napier published a description of how to make and use his rods only in the year of his death, and, in hindsight, they never quite scaled the heights of his other great invention. Certainly, Napier’s bones were better than logarithms in at least one sense, in that they allowed for precise multiplications of whole numbers rather. Yet as embodied in the slide rule, logarithms were astonishingly quick to use — as quick as an electronic pocket calculator, if somewhat less precise — and so the fate of Napier’s bones was sealed.

1.
Read, John. “Scottish Alchemy in the Seventeenth Century”. Chymia 1 (1948): 139-151. https://doi.org/10.2307/27757120.

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2.
Molland, George. “Napier, John, of Merchiston (1550–1617), Mathematician”. Oxford Dictionary of National Biography.

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3.
Bruderer, Herbert. “How Does One Multiply With Napier’s Rods?”. Accessed November 21, 2023.

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Shady Char­ac­ters advent calendar 2023: Chinese counting rods

Welcome to day two of the first ever Shady Characters advent calendar! I’m counting down to Christmas by way of a collection of beautiful, clever, important, and/or outright odd calculators and calculating devices. Some come from the pages of Empire of the Sum, some are part of my Calculator of the Day series, and some will be new to the blog. I won’t manage twenty-four posts, but I do hope to hit at least one every other day. I hope you enjoy the series!

When it comes to calculating devices, China is synonymous with the abacus. But this is a comparatively recent development. For hundreds of years before the abacus appeared, numerate Chinese people used a different method of counting and calculating.

Pascal's triangle, shown using Chinese rod numerals
This table, rotated 90° from its original layout, was published in Japan in 1712. Rather than the traditional counting rod board, this uses rod numerals to show the mathematical construction we call “Pascal’s triangle”, in which each number inside the triangle is the sum of the two numbers above it and above and to its right. (Image courtesy of the National Diet Library.)

Counting rods (called ‘籌’, or chóu) were a system in which short bamboo rods were placed on a grid similar to a chessboard, with the rightmost column holding units, the column to its left tens, the column to the left of that hundreds, and so on. Thus, each row represented a complete number and each column represented a single digit, with the user taking as many rows as necessary to work through their mathematical problem.1 Rods even supported negative numbers, with rods standing for negative numbers marked with a black dot and those for positive numbers with a red dot.2

The rods inspired a written notation called rod numerals that boasted an easily-understood visual construction and a robust place-value system (neither of which could be said of the contemporary Roman numerals), and which even supported the concept of zero (which would not arrive in the West until the thirteenth century). By the end of the fifth century, China had used rod numerals to calculate pi to seven decimal places — a feat that would not be replicated elsewhere in the world until a thousand years later. Indeed, rods and rod numerals were so useful that they made their way into Korea and Japan, as would the abacus in years to come.3

A <i>suanpan</i>, or Chinese abacus
A suanpan, or Chinese abacus. As was common in China, the upper and lower portions each contain an extra bead to make certain calculations easier. Conversely, Japanese abacuses, or sorobans, normally had only the required 4 and 1 beads. Another oddity here: this abacus, courtesy of Yale’s Peabody Museum, has been photographed upside down — the section with fewer beads is normally positioned above that with more beads. (Image courtesy of the Peabody Museum. Gift of Thomas G. Cary, 1877. )

Of course, rods and rod numerals have been thoroughly eclipsed by that later device. Yet the abacus contains echoes of the earlier system: each of an abacus’s columns, or wires, represents the digits 0 through 9, just like counting rods, and each column represents a new power of ten — again, just like counting rods.2 Moreover, most Chinese and Japanese abacuses split their rods into upper and lower sections, where the lower section’s four beads represent the values 1 through 4 and a single upper bead adds 5 to total 9. This mirrors rod numerals, where, for each digit, values 1 through 4 were represented by vertical rods (||||) and 5 was represented by a single, additional horizontal rod (–).

Comprehensive, portable, and easy to learn: counting rods and rod numerals were everything that their successor would turn out to be, lacking only, perhaps, in a little outright speed. They deserve better than to be forgotten.

1.
O’Connor, J J, and E F Robertson. “Chinese Numerals”. School of Mathematics and Statistics, University of St Andrews, 2004.

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2.
Schwartz, Randy K. “A Classic from China: The Nine Chapters - Numbers and Units”. Mathematical Association of America. Accessed December 3, 2023.

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3.
Swetz, Frank J. “Reflections on Chinese Numeration Systems: What Are Rod Numerals?”. Mathematical Association of America.

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Shady Characters advent calendar 2023: the tally stick

Welcome to day one of the first ever Shady Characters advent calendar! This is all a little experimental, but I plan to count down to Christmas by way of a collection of beautiful, clever, important, and/or outright odd calculators and calculating devices. Some come from the pages of Empire of the Sum, some are part of my Calculator of the Day series, and some will be new to the blog. I won’t manage twenty-four posts, but I do hope to hit at least one every other day. I hope you enjoy the series!

Tally sticks are our oldest known mathematical artefacts. The oldest of all, the “Lebombo Bone” from South Africa, is more than forty thousand years old.1 The principle behind the tally stick is simple: take a stick, carve a notch or a mark in it to add 1 to some ongoing count, and you have a portable, permanent record of that cumulative value.2

A wooden stick with notches carved along its top edge, writing inscribed on its front face, and the back face split off to form a separate piece.
A medieval English tally stick, split to record a debt and the debtor. It relates to a debt owed to the dean of Preston Candover in Hampshire of a sum of £2 13s 4d. (CC BY-SA 2.0 image courtesy of Hampshire Museums.)

But later, and in Britain especially, tally sticks took on another use: they were used to record debts. A stick of willow was inscribed with the name of the debtor and scored to record the amount. It was then split along its length, so that the two halves matched only each other. The half with the name of the debtor was kept by the creditor, and the other half was kept by the debtor. The “stock” — the part that identified the debtor and amount of their debt — became a form of currency, which could be exchanged in lieu of money. The new owner of the stock could then, at any time, demand settlement of the debt or trade it on to yet another owner.3

The government treasury, or Exchequer, used tally sticks to record debts owed to the state until as late as 1826, when the practice was replaced by written ledgers. Eight years later, when the last two cartloads of stocks were to be burned, janitors at the Houses of Parliament set about doing so in a furnace in the building’s basement — leading to a fire that very nearly destroyed the entire parliamentary estate.4

From recording prehistoric counts to managing the finances of the British state and immolating its parliament, the simple tally stick lived an eventful life.

1.
D’Errico, Francesco, Lucinda Backwell, Paola Villa, Ilaria Degano, Jeannette J. Lucejko, Marion K. Bamford, Thomas F. G. Higham, Maria Perla Colombini, and Peter B. Beaumont. “Early Evidence of San Material Culture Represented by Organic Artifacts from Border Cave, South Africa”. Proceedings of the National Academy of Sciences 109, no. 33 (August 14, 2012): 13214-13219. https://doi.org/10.1073/PNAS.1204213109.

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2.
Menninger, Karl. Number words and number symbols: a cultural history of numbers. Cambridge: MIT Press, 1969.

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3.
Harford, Tim. “What tally sticks tell us about how money works”. BBC News, sec. Business.

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4.
“Tally Sticks”. UK Parliament. Accessed December 1, 2023.

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We have audiobook winners!

The sun rises behind a pocket calculator, whose display reads "07734"
The cover of Empire of the Sum.

Congratulations to Piotr, William, Roslyn and Ian, winners of the Empire of the Sum audiobook giveaway! Their names were picked at random from the set of all entrants who replied to the original post about the competition.* Thank you all for taking part!

If you won, congratulations! Look out for an email from me with a code to access your audio copy of Empire. If not, don’t despair — Empire is available at audiobooks.com and many other audiobook purveyors. If you do listen to it, drop me a line, leave a comment, or post a review to let others know what you thought of it!

*
In more detail: I arranged the names of all entrants in a text file, then used random.org to pick two numbers between 1 and the total number of lines in that file. ↢

Win the Empire of the Sum audio book!

The Empire of the Sum audio book is out now, narrated by Elliot Fitzpatrick, and this is your chance to win one of four (four!) free copies at audiobooks.com. To enter, leave a comment on this post with a valid email address so that I can contact you in the event that you win.

The sun rises behind a pocket calculator, whose display reads "07734"
The cover of Empire of the Sum.

To receive your prize, you’ll need an account at audiobooks.com. audiobooks.co.uk also works, but I can’t guarantee that non-US, non-UK winners will be able to redeem their download code — although equally, I have no evidence to say they won’t be able to redeem them, either.

The con­test will close at noon UK time on Sunday 19th November 2023, so make sure you enter be­fore then. After that I’ll pick four win­ners from the list of all unique entrants, and I’ll get in touch with a code to access your copy via audiobooks.com. See below for terms and conditions, and good luck!

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