The mathematical patterns could just be coincidence, but they could also show the numerical sophistication of our ancient ancestors. The numbers 48 and 60 are 4 × 12 and 5 × 12 respectively, hinting that the people who made the scratches had a number system built around the number 12 (rather than 10, as we use today).#6496•
Base-60 may initially seem complicated compared to base-10, but it gave the Babylonians a mathematical edge. The number 60 is a superior highly composite number, meaning that it has many factors—it can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. This makes it easy to work with, particularly when writing fractions.#6481•
As 60 is a superior highly composite number, there are more fractions that can be expressed nicely in base-60 than in base-10.#6474•
Much of this is collated in the Rhind* papyrus, a manuscript written by a scribe called Ahmes. It is the oldest surviving mathematics textbook we know of and has this extraordinary opening: "Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets." Ahmes wrote the manuscript in around 1550 BCE and says that he used texts from around 2000 BCE to put it together.
That the mathematics it contains could be at least four thousand years old is hard to fully appreciate, especially considering that so much of what it contains resembles mathematics as we know it today.#6500•
The textbook contains eighty-four mathematical problems and ways to solve them. Six of the problems are about calculating the slope of a pyramid from its height and width using ideas akin to trigonometry. Mathematics is shaped by the people who develop it, so it is no surprise that Egyptian mathematicians were interested in the mathematics of pyramids when the pharaohs were so obsessed with building them.
But mathematical ideas are also universal.#6478•
The Maya also had another number system that was less decorative and more practical. It used two symbols: a dot and a bar. The dot represented 1 and the bar 5. Rather than being built around 10 or 60, like decimal and sexagesimal number systems, the Maya number system was vigesimal, meaning that it was built around the number 20.#6476•
The Maya made incredibly accurate measurements of the movements of the moon and stars: for instance, they calculated that 149 lunar months lasted 4,400 days; in our notation, this results in a lunar month of 29.5302 days, and today, we have it as 29.5306. Similarly, they worked out the length of the year as 365.242 days; today, we put this at 365.242198 days.#6502•
Chapter 2
The philosophy contained in The Book of Changes is linked to the concept of yin and yang that permeates ancient Chinese culture and says that two complementary halves must come together to produce wholeness. Yin comes from the word for the shady side of a hill and yang for the sunny side. Yin and yang are said to form the basis of all things, including human beings.
It was thought that the dynamics of the world could be understood through this lens with the help of The Book of Changes.
In the hexagrams, broken lines represent yin and solid ones yang, and the chart displays the sixty-four ways in which the two symbols can be combined in groups of six.
For hundreds of years, the great and the good consulted The Book of Changes to help them make decisions and understand their purpose in life, and the book took on such importance that all subjects had to embrace it.#6486•
Binary, in The Book of Changes, was deeply rooted in the philosophy of yin and yang, and Leibniz's binary was deeply rooted in Christianity. Nevertheless, the resulting mathematics was universal and clearly compatible with both Chinese and European culture—and with many others too. Binary mathematics also appears in the Rhind papyrus, in the mathematics of India in the second century BCE and, at least three hundred years before Leibniz was born, in the counting system of the Mangareva people of French Polynesia.#6501•
The origins may have been different, but the binary fundamentals were the same. Mathematics is often intertwined with religion, politics, culture and identity—it is performed by people, after all, so it's hard to imagine it any other way. However, as the case of binary shows, there are many ways to reach a mathematical idea.#6477•
Chapter 3
As Plato wrote in his Republic, "geometry will draw the soul towards truth." Mathematics was about uncovering "knowledge of the eternal."#6498•
Chapter 4
The gnomon, or shadow stick, was used across the ancient world from about 3500 BCE to estimate the time of day. The slim bar cast a shadow, and its length was then measured. A sundial is a gnomon with a display added, and these were located in public places so people could check the time throughout the day.
If accurately made, sundials can tell the time down to the minute, but they have an obvious drawback—they don't work at night, or when it's cloudy.#6470•