The Number 360: The Mathematical Key to the Babylonian Sky

360 degrees: the oldest unit of measurement still in use?

A division born of celestial observation

Open a protractor, look at a compass dial, or consult a map: everywhere, the circle is divided into 360 degrees. This number seems self-evident, as if it were written into the nature of things. Yet nothing in pure geometry requires 360 rather than 100 or 1000. This number is the fruit of a human choice, more than three millennia old, and that choice was dictated by the sky.

The starting question is almost childlike: the Earth rotates on its axis in 24 hours, thus defining the day. It revolves around the Sun in about 365 days, describing an almost circular orbit. And a circle is 360 degrees, or almost exactly 1 degree per day. Is there a link between these three observations? The answer is yes, but in an unexpected sense: it is not nature that chose 360, it is humanity that chose 360 by drawing inspiration from nature.

The Babylonians, pioneers of mathematical astronomy

A sexagesimal system in the service of the sky

It all begins in Mesopotamia (historical region corresponding to present-day Iraq, cradle of mathematical astronomy), between the Tigris and the Euphrates, in what is now Iraq. As early as the 2nd millennium BCE, Babylonian scribes recorded their observations of the sky on clay tablets with remarkable precision. They used a sexagesimal (counting system based on base 60, still used for hours, minutes, and degrees) numeral system, that is, base 60. This is why we still count 60 seconds in a minute, 60 minutes in an hour, and 60 arcminutes in a degree.

These astronomer-mathematicians observed night after night the progression of the Sun across the celestial vault. From one night to the next, the Sun moves about 1 degree among the fixed stars, along a great circle that the Greeks would later call the ecliptic (apparent annual trajectory of the Sun on the celestial sphere, crossing the 12 constellations of the zodiac). The Babylonians found that the Sun returned to its initial position after about 360 days, a convenient approximation of the real year (365.25 days). It was then natural, for minds accustomed to base 60 and eager to divide this great perfect celestial circle, to choose 360: a number both close to the year and extraordinarily rich in divisors.

360: an arithmetically perfect number

A wealth of divisors

The choice of 360 is not due solely to the sky. It also owes much to mathematics. The number 360 has 24 integer divisors:

• 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.

This arithmetic richness is crucial for astronomers and engineers who, without calculators, had to divide circles into equal portions by hand. Dividing a circle into 2, 3, 4, 5, 6, 8, 9, 10, or 12 equal parts always gives a whole number of degrees. No other number of comparable size offers such computational convenience.

From Babylon to Greece: the transmission of knowledge

Hipparchus and Ptolemy, heirs and transmitters

It was the Greek astronomers who inherited and formalized the Babylonian system. Hipparchus of Nicaea (c. 190–120 BCE), considered the founder of positional astronomy, explicitly used the division of the circle into 360 degrees to map the stars and measure celestial positions. He was also the first to systematically subdivide these degrees into 60 arcminutes and each minute into 60 arcseconds, definitively consolidating the Babylonian sexagesimal legacy.

Claudius Ptolemy (c. 100–170), in his major work the Almagest, took up and amplified this framework. The Almagest remained the reference of Western and Arab astronomy until the 16th century, thus ensuring the transmission of the Babylonian degree to all subsequent scientific civilization. It is this uninterrupted thread that explains why a French engineer, a Japanese navigator, and a Brazilian architect all today use exactly the same angle of 360 degrees.

Axial rotation and orbital revolution: two independent movements

Two movements of distinct origins

Let us now return to the initial physical question. Are the day (Earth's rotation on its axis, about 24 hours) and the year (revolution around the Sun, 365.25 days) physically linked? The answer from astrophysicists is clear: no.

These two movements have distinct origins:

• The Earth's axial rotation is a legacy of the formation of the solar system, 4.56 billion years ago. When the protoplanetary disk collapsed under its own gravity, the conservation of angular momentum (physical quantity that measures the amount of rotation of a system, and which is conserved in the absence of external torque) set the proto-Earth into rotation. Since then, this rotation has been slowing very slowly under the effect of tidal friction exerted by the Moon: in the age of the dinosaurs, a day lasted only about 23 hours.
• The orbital revolution is entirely governed by the Earth-Sun distance and the mass of the Sun, in accordance with Johannes Kepler's third law (1571–1630): \( T^2 \propto a^3 \), where \( T \) is the orbital period and \( a \) the semi-major axis of the ellipse. If the Earth were closer to the Sun, its year would be shorter, but its day would remain unchanged.

The fact that the year contains about 365 times the duration of a day is therefore a pure geographic contingency, linked to the accidental distance at which the Earth formed around its star. There is no physical mechanism that synchronizes these two time scales.

The real coincidence: 360 versus 365

A known but accepted discrepancy

Here is the crucial point that intuition can miss. The year does not last 360 days, but 365.25 days. The degree therefore does not correspond exactly to the Sun's daily displacement. In reality, the Sun moves each day by \( \frac{360°}{365.25} \approx 0.9856° \), that is, almost, but not quite, 1 degree.

The Babylonians knew this. Their most precise astronomical tablets, such as the MUL.APIN (collection of Babylonian tablets describing planetary movements with great precision, dating from the 1st millennium BCE) series (c. 1000 BCE), already carefully distinguished the 365-day year from the idealized 360-day year. They deliberately chose 360 as the base of their angular system, accepting a slight approximation, because the arithmetic advantages of this number far outweighed the disadvantage of a discrepancy of a few days.

Comparison of systems of angular division of the circle

The table below compares the main divisions of the circle used throughout history and their numerical bases.

N.B.:
The radian, the International System (SI) unit for angles, is defined as the ratio of arc length to the radius of the circle. A full turn equals exactly \( 2\pi \) radians, which makes the formulas of physics and mathematical analysis more elegant than in any other angular unit. But in daily practice, the Babylonian degree reigns unchallenged.

360: A 3,000-year-old legacy

A convention still alive

The Earth turns, the seasons return, and our circles have 360 degrees because astronomers, more than 3,000 years ago, decided that geometry would follow the rhythm of the stars. It is not nature that imposed this number, it is human intelligence that chose it by reading nature. The scribes of Babylon, armed with reeds to engrave fresh clay and eyes fixed on the night sky of Mesopotamia, chose 360 because this number reconciled two contradictory requirements: staying true to the rhythm of the sky and lending itself easily to hand calculation.