All right. Now, we start getting the complications. The technical term is inequality. What inequality means is non-uniformity in motion. One would like it if the Sun for example, moved exactly the same amount among the stars every day. But it appears not to. In particular, now turning to the slide itself, in some parts of the zodiac, it moves faster, angularly, in some parts slower, so that you get 94 and a half days from Vernal Equinox to the Summer Solstice and 92 and a half days from the Summer Solstice to Autumnal Equinox where we are now for a couple more weeks.

So the Sun does not move uniformly. It does not appear to move uniformly. Whether it moves uniformly or not is a separate question. It appears to move non-uniformly. And that's called an inequality because it's a violation of equal motion. Equal angular motion, associated with that, but it's in parentheses because it doesn't matter that much to us.

The length of a solar day from one noon to the next, deviates from mean solar time by as much as 16 minutes over the course of a year. So some days from noon to noon, you can tell what noon is by putting up a gnomon and seeing where the shadow is longest.

The moment the shadow was longest. The actual time from noon to noon varies from day to day. That was recognized well before Ptolemy. They had figured out exactly why Ptolemy explains it properly, part of it is the obliquity of the ecliptic, the fact that the Sun is on an angle.

Another part of it is the sun isn't really moving uniformly. But there's a correction. It's called the equation of time, to give you local time transferred to mean solar time. Mean solar time consists of a year, which for Ptolemy was 365 days plus a quarter of a day, minus one three hundredths of a day.

And that's off by a few minutes unfortunately. But that's getting reasonably close. Divide that now. Divide the whole year by 365 a quarter etc. And that gives you mean solar time. Okay, per day. The number of degrees per day, and it's always referenced as mean solar time. That's going to be, that is still our unit of time.

That's mean solar time up there. It turns, there's a nice little trick. If, now I'll revert to, well I don't have to revert to modern form. As the Sun goes around the Earth in the course of a year, it manages to add a day, as far as the number of times at any spot, you see the Sun over it.

Can everybody see that? The Sun's going around the Earth is turning every day. But by the time it gets back, it will have gone once around the Earth and one more day. So, that means Sidereal time, which is the time now fundamental, or for most of this course will be fundamental.

Is 24 hours, 56 minutes and I think it's four seconds, I believe. That's 366 and a quarter, minus one three hundredths, divided by 365 and one quarter minus 300. Flip that and multiply it by 24 and you get the Sidereal day. That's the time any one star crosses the meridian, and that is as regular as it can be.

Okay, that's the basis for time and that's the way you end up defining mean solar time, most conveniently locally rather than spending a whole year. Is that halfway clear? Okay, so we've got the Sun speeding up and slowing down. Second thing, the moon speeds up and slows down during the course of its 27 plus day orbit, just the way the Sun does.

So it's faster in some places, slower in others. Also, the moon is at a slight angle compared to the ecliptic. And eclipses can only happen when the moon is right on the ecliptic, in line with the Sun and the Earth. Okay? Now, it turns out the line of intersection between the lunar orbit and the Earth Sun orbit, it's called the line of nodes.

It's not stationary. It slowly regresses, taking 18 years to go to full circle. Similarly, the point of the moon that's furthest from the Earth, the apogee moves forward taking nine years to move forward. Okay, so in addition to the moon speeding up and slowing down, the whole orbit can be thought of as moving in space.

That was fully known to the Babylonians, they managed to figure that out and were able to predict solar and lunar eclipses to a very high accuracy, discovering that every 18 years the pattern repeats itself, all the things come back. It's not quite exactly 18 years, it's very close.

So that's one of the things that the Babylonians are legitimately famous for. They figured out the pattern of solar eclipses and lunar eclipses, which is a very complicated story. I'm gonna make that story more complicated in just a moment. Let me go on to the planets though. Just like the sun and just like the moon, each planet goes faster in part of the zodiac than it does in other parts of the zodiac.

The one I've thrown up there is the case of Mars, the angular velocity is 40% faster when it's in Capricorn than when it's on the opposite side in Cancer, and that's the pattern that repeats every 687 days. So that actually has a name from time immemorial. It's called the First Inequality of the Planets.

It's common to the moon, the Sun, and the planets. The speeding up and slowing down, but I repeat, the keyword in the title up there, observed. That is, it appears to be speeding up and slowing down, okay. Whether it is, we will get to in a little bit.