Probably the easiest description to understand is that the equation of time is the difference between the time told on a sundial, and the time told by a clock; although, this naturally raises the question of why there’s a difference — to say nothing of the question of why anyone would care.
A little thought experiment to get us in the right frame of mind to understand the beauty of the equation of time complication. Imagine that all the artificial lights that fog the evening sky are gone, and that the only source of light in your world, other than the faint flicker of whatever fire you can kindle, is the sun. Darkness now is something unimaginably terrifying — impossible to escape, full of glowing eyes and mortal peril. In such a world, the rising of the sun is like a delivery from death in a way the modern mind can hardly imagine — you’d pay attention too.
The Analemma Dilemma
Mankind has studied the heavens for a long time, patiently charting/calculating the movement of the stars, exhausting intellect and imagination in contemplating the very nature of objects a veritable eternity away. Sun, moon earth and stars – which revolves around what? What are they anyway? What is this “nothing” upon which the earth supposedly hangs?
By the end of the 17th century though, two things had become clear. The first was that the earth orbited around the sun; not in a perfect circle, but in an ellipse. The second was that the earth’s axis wasn’t perfectly vertical — in fact, it’s significantly tilted, about 23 degrees and seven minutes of arc.
The seasons are mostly due to the tilt of the earth’s axis. In the northern hemisphere, when the North Pole is tipped towards the sun, days are longer and the sun is higher in the sky; when the North Pole is tipped away from the sun (six months later), days are shorter and the sun is lower in the sky.
If you were to measure the height of the sun above the horizon every day at exactly noon on the clock, you’d see that the sun’s height changes a little every day — highest at the summer solstice, lowest at the winter solstice.
So far so good. But you’d notice something else, too — the moment when the sun is at the highest point in the sky each day is not always exactly noon! In fact, the sun’s culmination, or the time it reaches its zenith, matches the clock, disturbingly enough, only four days a year.
Annoyed by this, you decide to get to the bottom of the mystery, and so every day at noon on your clock, you take a picture of the sun. At the end of the year, you superimpose all the photos and you notice something very weird — over a year, the positions of the sun in the sky, at noon, trace out a strange-looking figure eight. What the hell??
This wacky figure eight in the sky is called an analemma, and the fact that the sun is sometimes ahead of where it should be at noon, and at other times behind where it should be at noon, is what the equation of time hand on a watch illustrates.
The reason for the discrepancy between clock time and the position of the sun in the sky — which, naturally, is the time a sundial’s going to show — is because of the earth’s tilted axis. Because its orbit is elliptical, the sun appears to speed up and slow down over the course of a year.
An object orbiting along an elliptical path is going to speed up and then slow down as it zips along between the perihelion (the point closest to the sun) and the aphelion (the point furthest from the sun). This speeding up and slowing down contributes a variation in sun time with a one-year cycle to the equation of time. But the earth’s tilted axis not only makes the sun appear sometimes higher and sometimes lower in the sky — because the tilt makes the sun seem to wobble not only up and down, but also back and forth, it also contributes to the equation of time — but in a twice-yearly cycle.
The equation of time, in either a watch or a clock, can be indicated in various ways — the most common way is to have a sector on the dial with a hand that shows how much you would have to add or subtract to the mean solar time to get the apparent solar time, or sundial time. The hand waggles back and forth as it’s moved by gears attached to a metal finger that traces out the contours on a vaguely kidney-shaped cam that rotates once a year, the outline of the cam corresponding to the analemma.
The equation of time is an unusual complication to begin with, even in its simplest form; a more complex and even less often seen variation is the equation of time marchante, in which there are two minute hands on the dial — one for the mean time and one for the apparent solar time (in other words, the equation of time hand). Over the course of a year, the EOT hand slowly overtakes and then slowly falls behind the minute hand as the sun overhead gradually catches up with and then falls behind clock time.
Historically, the equation of time was (and still is) a rarity. Most often, it’s found in clocks and rarely in pocket watches. Such watches were manufactured by a veritable who’s who of horological history: Thomas Mudge in the 18th century, and luminaries such as Leroy, Breguet and Berthoud in the 19th century, manufactured equation of time watches by request for only the most illustrious of clients. Of course, the most famous of all watches employing the equation of time is the missing Breguet ‘Marie Antoinette’ — the Holy Grail of vintage horology. For much of horological history, if an equation of time watch was in your waistcoat pocket, you had to be either a merchant prince or a real prince (or princess).