Why Time Needs Measuring

Imagine you are a hunter-gatherer living some forty thousand years ago, and you want to meet your cousins at the river when the salmon run starts. You cannot send a text, there is no calendar on the wall, and nobody has invented a number for "next month". What you have is the sky. The Sun comes up and goes down. The Moon grows fat, thins away and returns. The seasons swing round and the cold comes back. These three rhythms are the oldest clocks there are, and for almost the whole of human history they were the only ones. This course follows the long story of how people learned to count them, argue about them, build machines to imitate them, and finally replace them with the vibrations of atoms.

Three Natural Clocks

Each natural clock comes from a different motion of the Earth or the Moon, and that is the root of every calendar headache in the chapters to come.

The Earth spinning, the Moon circling and the Earth circling the Sun are three separate motions. Nothing in physics makes one of them a tidy multiple of another. As the next section shows, the numbers do not fit.

Why They Do Not Divide Evenly

How many months are in a year? If the answer were a whole number, calendar-making would be easy. Divide the year by the month:

A year contains 12.368 Moon cycles: twelve and a bit. Twelve months add up to 354.367 days, which is 10.875 days short of a year. Thirteen months overshoot by 18.655 days. So a calendar built purely on the Moon slides backwards through the seasons by nearly eleven days a year, and a calendar built purely on the Sun ignores the Moon completely. Every calendar in history is a compromise between these two facts, and the compromises fill chapters 3 and 4.

The day has the same trouble with the year. The year is 365.2422 days, so if you count days you must carry that extra 0.2422 of a day (nearly six hours) somewhere. A calendar of exactly 365 days loses about a quarter of a day each year. After four years it is nearly a whole day (0.97 of a day) behind the seasons.

A Closer Look at the Day

Even "a day" is less simple than it sounds. There are two sensible ways to measure one, and they give different answers.

Why the difference? While the Earth spins once, it also moves a little along its orbit around the Sun. After one full turn relative to the stars the Earth has to turn a little further for the Sun to be back at its highest. Over a whole year, this adds up to exactly one extra turn of the Earth, so a year contains one more sidereal day than solar days: 366.2422 sidereal days, against 365.2422 solar days. The sidereal day is the solar day scaled by that ratio:

The sidereal day is 86,164.09 seconds, or 23 h 56 min 4.09 s, which is 235.91 seconds (about 3 min 56 s) shorter than the solar day. That is why a given star rises about four minutes earlier each night, and why different constellations dominate the sky in different seasons.

Neither day is perfectly constant. The solar day varies slightly in length through the year (chapter 2 explains the equation of time), and tides slowly drag on the Earth's spin so that the day is getting longer by roughly a couple of milliseconds per century. For now, take the "24 hours" day as an average.

The Oldest Records: Notches on Bone

How early did people count these cycles? We cannot know what they said or thought, but some objects look very much like tallies. The most famous are two bones from Africa:

• The Lebombo bone, from a cave in the Lebombo mountains on the border of South Africa and Eswatini, is a piece of baboon leg bone about 43,000 years old, with about 29 clear notches cut in it. The bone is broken at one end, so there may once have been more. A count near 29 is close to a lunar month, which is why it is sometimes called a lunar tally.
• The Ishango bone, found in what is now the Democratic Republic of the Congo, is roughly 20,000 years old and carries groups of notches in three columns. It has been read as a tally, as a lunar record and even as a simple arithmetic aid.

Be careful with these claims. Marks on bone could be a hunting tally, a trade record or decoration, and nobody can prove which. What is safe to say is this: tens of thousands of years ago, people made patterned marks in sets that look like counting, and counting the Moon is one reasonable explanation. A tally keeps one thing the memory cannot: a number you can check tomorrow.

Tally Drift: Why Counting Is Not Enough

Suppose a bone-carver cuts one notch per day and starts a new column at each new Moon. They must decide how long a month is. Real months last 29 or 30 days, never both at once. Counting every month as 29 days makes the tally lag the Moon. Counting 30 makes it run ahead. How quickly?

A fixed 29-day month falls 6.4 days behind the Moon in a year. A fixed 30-day month runs 5.6 days ahead. Even the best fixed guess, 29.5 days (alternating 29 and 30), is still 0.367 days out after twelve months and a full day out after about 33 months. A tiny error in a repeated count does not stay tiny: it adds up. This is the key idea of the whole course, and we will meet it again with calendars, with pendulum clocks, with ships at sea and with computers. A timekeeper is judged less by how well it starts than by how its error grows.

What Makes a Good Clock?

We can now ask what we mean by a "good" clock. People use three words loosely, but they mean different things. Definitions vary a little between fields. These are the working meanings used in this course.

A very stable clock is the most useful, even if it is inaccurate, because a steady error can be measured once and corrected for ever. A stable clock that gains exactly 2 seconds a day can be set right by subtracting 2 seconds every day. An unstable clock cannot be corrected in advance. This is why later chapters spend so much effort on making the rate steady (pendulums, balance springs, quartz, atoms) rather than on setting the starting time.

Errors are often quoted as a fraction. A clock that gains 1 second in a day is wrong by 1 part in 86,400. Engineers write this as parts per million (ppm):

A clock only 1 second a day out looks wonderful, yet it is 6 minutes a year out, and 10 seconds a day means an hour a year. For navigation, science and the internet, these small errors are not small at all. By the end of the course you will see clocks that are wrong by less than one second in many millions of years.