[My wife is a compulsive counter. If we are out on a walk and see a flock of turkeys in a field, she’ll begin one, two, three. The flock will move and she’ll start again, one, to, three … SB SM]
Lapham’s Quarterly embodies the belief that history is the root of all education, scientific and literary as well as political and economic. Each issue addresses a topic of current interest and concern—war, religion, money, medicine, nature, crime—by bringing up to the microphone of the present the advice and counsel of the past.
The Early History of Counting
How ancient peoples learned to keep a tally.
WEDNESDAY, AUGUST 23, 2023
Figuring out when humans began to count systematically, with purpose, is not easy. Our first real clues are a handful of curious, carved bones dating from the final few millennia of the three-million-year expanse of the Old Stone Age, or Paleolithic era. Those bones are humanity’s first pocket calculators: For the prehistoric humans who carved them, they were mathematical notebooks and counting aids rolled into one. For the anthropologists who unearthed them thousands of years later, they were proof that our ability to count had manifested itself no later than 40,000 years ago.In 1973, while excavating a cave in the Lebombo Mountains, near South Africa’s border with Swaziland, Peter Beaumont found a small, broken bone with twenty-nine notches carved across it. The so-called Border Cave had been known to archaeologists since 1934, but the discovery during World War II of skeletal remains dating to the Middle Stone Age heralded a site of rare importance. It was not until Beaumont’s dig in the 1970s, however, that the cave gave up its most significant treasure: the earliest known tally stick, in the form of a notched, three-inch long baboon fibula.On the face of it, the numerical instrument known as the tally stick is exceedingly mundane. Used since before recorded history—still used, in fact, by some cultures—to mark the passing days, or to account for goods or monies given or received, most tally sticks are no more than wooden rods incised with notches along their length. They help their users to count, to remember, and to transfer ownership. All of which is reminiscent of writing, except that writing did not arrive until a scant 5,000 years ago—and so, when the Lebombo bone was determined to be some 42,000 years old, it instantly became one of the most intriguing archaeological artifacts ever found. Not only does it put a date on when Homo sapiens started counting, it also marks the point at which we began to delegate our memories to external devices, thereby unburdening our minds so that they might be used for something else instead. Writing in 1776, the German historian Justus Möser knew nothing of the Lebombo bone, but his musings on tally sticks in general are strikingly apposite:
The notched tally stick itself testifies to the intelligence of our ancestors. No invention is simpler and yet more significant than this.
It is not clear what quantity the twenty-nine notches carved into the Border Cave’s baboon fibula represents. It is a number, that much is known: had the bone been purely decorative, the notches would have been added all at once, but four different tools were used over time to add to the count. As such, the Lebombo bone is likely to be the earliest mathematical device ever found. (Sadly, it is too great a leap to call it the earliest known pocket calculator. Humans started wearing clothes around 170,000 years ago, but pockets themselves are probably no more than a few thousand years old.)If the Lebombo bone answers the question, at least partly, of when humans learned to count, it leaves another one unanswered: How did they learn to do so?
Counting, fundamentally, is the act of assigning distinct labels to each member of a group of similar things to convey either the size of that group or the position of individual items within it. The first type of counting yields cardinal numbers such as “one,” “two,” and “three”; the second gives ordinals such as “first,” “second,” and “third.”
At first, our hominid ancestors probably did not count very high. Many body parts present themselves in pairs—arms, hands, eyes, ears, and so on—thereby leading to an innate familiarity with the concept of a pair and, by extension, the numbers 1 and 2. But when those hominids regarded the wider world, they did not yet find a need to count much higher. One wolf is manageable; two wolves are a challenge; any more than that and time spent counting wolves is better spent making oneself scarce. The result is that the very smallest whole numbers have a special place in human culture, and especially in language. English, for instance, has a host of specialized terms centered around twoness: a brace of pheasants; a team of horses; a yoke of oxen; a pair of, well, anything. An ancient Greek could employ specific plurals to distinguish between groups of one, two, and many friends (ho philos, to philo, and hoi philoi). In Latin, the numbers 1 to 4 get special treatment, much as “one” and “two” correspond to “first” and “second,” while “three” and “four” correspond directly with “third” and “fourth.” The Romans extended that special treatment into their day-to-day lives: after their first four sons, a Roman family would typically name the rest by number (Quintus, Sextus, Septimus, and so forth), and only the first four months of the early Roman calendar had proper names. Even tally marks, the age-old “five-barred gate” used to score card games or track rounds of drinks, speaks of a deep-seated need to keep things simple.
Counting in the prehistoric world would have been intimately bound to the actual, not the abstract. Some languages still bear traces of this: a speaker of Fijian may say doko to mean “one hundred mulberry bushes,” but also koro to mean “one hundred coconuts.” Germans will talk about a Faden, meaning a length of thread about the same width as an adult’s outstretched arms. The Japanese count different kinds of things in different ways: there are separate sequences of cardinal numbers for books; for other bundles of paper such as magazines and newspapers; for cars, appliances, bicycles, and similar machines; for animals and demons; for long, thin objects such as pencils or rivers; for small, round objects; for people; and more.
Gradually, as our day-to-day lives took on more structure and sophistication, so, too, did our ability to count. When farming a herd of livestock, for example, keeping track of the number of one’s sheep or goats was of paramount importance, and as humans divided themselves more rigidly into groups of friends and foes, those who could count allies and enemies had an advantage over those who could not. Number words graduated from being labels for physical objects into abstract concepts that floated around in the mental ether until they were assigned to actual things.Even so, we still have no real idea how early humans started to count in the first place. Did they gesture? Speak? Gather pebbles in the correct amount? To form an educated guess, anthropologists have turned to those tribes and peoples isolated from the greater body of humanity, whether by accident of geography or deliberate seclusion. The conclusion they reached is simple. We learned to count with our fingers.
A 1913 survey of the number words used by several Native American tribes found that many of those words were related to “finger,” “thumb,” and “hand.” Counterintuitively, perhaps, despite the general possession of ten fingers per person, fewer than half of those tribes counted in multiples of ten. About a third used systems that revolved around the number 5, which was often referred to as “fingers finished,” “all finished,” “gone,” or “spent.” A further tenth of the tribes used vigesimal schemes based on the number 20 (“all hands and feet”), while a few contrarian outliers used 2-, 3-, and 4-based systems with less obvious connections to human anatomy.
Fifteen years earlier, a group of scientists from Cambridge, England, had made a series of visits to the islands of the Torres Straits, strung between Papua New Guinea to the north and Australia to the south. A.C. Haddon, the driving force behind the expeditions, recounted
There was another system of counting by commencing at the little finger of the left hand, kotodimura, then following on with the fourth finger, kotodimura gorngozinga (or quruzinger); middle finger, il get; index finger, klak-nětoi-gět; thumb, kabaget; wrist, perta or tiap; elbow joint, kudu; shoulder, zugukwoik; left nipple, susu madu; sternum, kosa, dadir; right nipple, susu madu, and ending with the little finger of the right hand.
In this way, Haddon said, starting on one side of the body and traversing over to the other, the islanders could count to nineteen. More recently, a math teacher named Glen Lean catalogued the number words for 883 of the 1,200 known languages from Papua New Guinea and Micronesia and found that the use of fingers for counting was foundational to many of those languages. Like the Torres Strait islanders, the Papua New Guineans then carried on to the forearm, elbow, eyes, nose, ears, and other body parts. A study of Yupno, a language indigenous to Papua New Guinea’s Finisterre Mountain range, recorded that Yupno men added their testicles and penis for good measure, allowing them to count to thirty-three using body parts alone. Hold my earliest attested beer, an ancient Sumerian might have said.
From the sixth millennium onward, the valley between the Tigris and the Euphrates Rivers—Mesopotamia, the ancient Greeks called it, the “land between rivers”—harbored one of the world’s earliest civilizations. Having mastered animal husbandry and the cultivation of crops, Mesopotamian farmers became the engine of a new agrarian economy. Almost from the beginning, it seems, they used small clay tokens an inch or so in size, hand-rolled into the shape of spheres, cones, disks, and other simple shapes, to keep records. Each shape stood for a fixed quantity of some good or other: A cone represented a small quantity of cereal, a sphere a larger amount, and a flat disk the largest. Ovoids were jars of oil; cylinders and rounded disks were farm animals; and so on.
Around 3300 bc, as Mesopotamia’s scattered farming communities began to coalesce into the patchwork of city-states called Sumer, their use of tokens became more sophisticated. At first, batches of tokens were wrapped in clay balls called bullae and marked with personal seals to create records of important transactions. Later, the surfaces of those bullae were impressed with the tokens to be sealed inside so that a bulla’s contents could be divined without having to break it open. Once it became apparent that the signs on the outside were as useful as the tokens on the inside, the tokens themselves became surplus to requirements—and the signs, says a theory first proposed by French-American archaeologist Denise Schmandt-Besserat, evolved into the distinctive angular form of cuneiform writing.
Cuneiform tablet with account of expenditures, Neo-Sumerian, c. 2043 BC. The Metropolitan Museum of Art, purchase, by exchange, 1911.Cuneiform tablets show that the Sumerians and their successors, the Akkadians and Babylonians, used sexagesimal numbers. That is, their numerical system was rooted in the number 60. Whereas decimal gives rise to round numbers such as 1, 10, and 100 (or 10 squared), the Sumerians counted in terms of 1, 60, 3,600 (or 60 squared), and so on. There are practical advantages to this, since 60 can be divided into whole numbers by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, but, as E. F. Robertson, late of St. Andrews University in Scotland, points out, it is rare for a culture to choose the base for its number system. More often, as illustrated by those Native American tribes, it naturally settles upon a base when it begins to count. Counting on five fingers leads to the quinary system, or base 5; two hands lead to decimal, or base 10; two hands and two feet to base 20, or vigesimal.
How, then, did the Sumerians land on base 60?The answer lies in the Sumerians’ tokens and bullae. Successive scholars have noted that the shapes made when tokens were pushed into the soft clay of a bulla appear to be very similar to the number symbols used on the earliest “proto-literate” clay tablets. That is, the shapes and the values of physical tokens seem to have carried over directly to the written sexagesimal numerals used by the earliest literate Sumerians. As such, the ancient Mesopotamians must have been counting in base 60 on their fingers long before they, or, indeed, anyone else on the planet, could set out numbers in writing.
The Mesopotamians’ unique counting method is thought to come from a mix of a duodecimal system that used the twelve finger joints of one hand and a quinary system that used the five fingers of the other. By pointing at one of the left hand’s twelve joints with one of the right hand’s five digits, or, perhaps, by counting to twelve with the thumb of one hand and recording multiples of twelve with the digits of the other, it is possible to represent any number from 1 to 60. However it worked, the Mesopotamians’ anatomical calculator was a thing of exceptional elegance, and the numbers they counted with it echo through history. It is no coincidence that a clock has twelve hours, an hour has sixty minutes, and a minute has sixty seconds.
Excerpted from Empire of the Sum: The Rise and Reign of the Pocket Calculatorby Keith Houston. Copyright © 2023 by Keith Houston. Used with permission of the publisher, W.W. Norton & Company, Inc. All rights reserved.
Keith Houston is the author of Empire of the Sum: The Rise and Reign of the Pocket Calculator, Shady Characters, and The Book. His writing has appeared in the New York Times, the Wall Street Journal, the Financial Times, Mental Floss, BBC Culture, and on Time.com. He lives in Birmingham, England.