Mathematical equations can be beautiful, or they can be ugly and messy. When we find a simple, elegant equation, we can rightly say it’s beautiful. Most people are familiar with the simple and elegant equation E = MC². It’s sometimes called Einstein’s equation and it expresses with admirable mathematical succinctness the relationship between energy, matter and the speed of light — a deep and astonishing fact of nature expressed in three letters.
Not so many people are acquainted with the equation on the right. It’s a humble formula, having nothing to do with the speed of light, atomic power plants, or thermonuclear bombs. It’s about pendulums. This equation expresses the relationship between the time it takes a pendulum to swing back and forth, the length of pendulum and the acceleration of a falling body due to gravity. If you recall the math you were taught in school, you say the Time it takes the pendulum to make one swing back and forth is equal to two times the value of pi, times the square root of the Length of the pendulum divided by the acceleration of gravity. The equation is associated with Galileo, and to some it’s as beautiful as the one about mass and energy.
According to one of Galileo’s students, Galileo was attending a religious service in Pisa when he noticed that a breeze caused very slight, very slow back-and-forth motion of a chandelier suspended in the great cathedral. Galileo’s mind was not focused on the sacred service being performed that day; instead he kept looking at the slow and gentle motion of the chandelier and he noticed that even though the breeze stopped and the back-and-forth distance traveled by the pendulum shrank, yet the time it took the chandelier to make the back-and-forth trip seemed to remain constant. There were no clocks back in 1582 — he’d invent one later — so he timed the swinging of the chandelier by the regular beating of the pulse in his wrist. He was right; no matter the distance traveled, the time it took was always the same.
Later, Galileo experimented with pendulums and discovered that the remarkably regular period of the pendulum (the uniform time it took to make a full back-and-forth sweep) was proportional to the square root of the length of the pendulum. The pendulum bob (the weight at the end of the pendulum) had no effect on the length of time or its regularity. The only things that mattered were the pendulum’s length and, of course, gravity that caused the pendulum to swing once it was released.
We don’t know what thought processes were going on in Galileo’s mind as he put together the different things he discovered about pendulums. He worked with proportions far more than we do nowadays when assembling data. So, let’s say that you, or Galileo, are interested in how long it takes a pendulum to make one complete swing back and forth, and all you know for certain is that it depends on the length of the pendulum and gravity.
Galileo knew something about gravity, too. You know the story about Galileo dropping different size canon balls from the Leaning Tower of Pisa to demonstrate that big ones and little ones hit the ground at the same time — that story is a fanciful myth. But he did know that all falling bodies accelerate at the same rate, increasing velocity at some number of seconds per second of fall. So when you let go of the pendulum bob, it will go faster and faster as it falls, then slow down more and more as it rises.
If you want to express time in, say, seconds, and you know that it’s the result of a mix of length and gravity, you have to compose an equation such that the units of length get removed and only seconds remain. If the acceleration of gravity is expressed as, say, centimeters per second per second, and the pendulum’s length is expressed in centimeters, then by dividing the pendulum’s length by the acceleration of gravity you get rid of the centimeters and are left with seconds per seconds. And if you take the square root of that, you re left simply with seconds. So the relationship between the time T and the length l and gravity g looks like the proportional formula on the left, below.
The two times pi, as in the in the equation at the top right, turns the proportional expression into a true equation — but that involves a mathematical maneuver that Galileo didn’t get around to. And we won’t either. (By the way, the portrait of Galileo is by the artist Domenico Robusti, also called Tintoretto, son of the more famous Tintoretto. Galileo was himself the son of the famous musician Vincenzo Galilei, a performer, composer and theorist of music. From his father, Galileo learned the mathematics of musical harmony and with that as a start it’s not surprising that when deep into the physics of the natural world he declared that the language of God’s creation was mathematics. )