 Okay, so I have to work out what happens when I double the point size of a word in terms of how much ink I need. Now obviously the amount of ink I need depends on the volume of ink. So it's the volume of ink that make up that letter. But when I double the point size, I'm not really changing the thickness of that letter as it's written on the page. I assume so. And so really what matters is the area. So if I have an area, so if I'm writing a letter A, is it why I don't design fonts? That's a terrible A. So I'm changing the length, which is the point size, that I'm also going to have to change the height. And so the area is going to change proportional to that length squared. So if I double the length, my area is going to be 2L all squared, which is going to be 4 times L squared. So if I double the length, I'm going to quadruple the area. So how much more ink? I need 4 times as much ink. Okay, I've got a scale model of a house. A scale model of a house where one centimeter is equal to one meter. Now I know that a centimeter is a hundredth of a meter. Take life in my own hands and try and draw a house. Now I'm going to have a length here, and a width, and a height. And when I make the real house, I'm going to have to make everything bigger. So that L is going to have to be 100 times bigger, that W is going to have to be 100 times bigger, and that H is going to have to be 100 times bigger. Now the question is, what do I need to calculate to figure out how much paint I'll need and how much wood I'll need? And they're actually different. So when we scale this up, all those surfaces are going to be covered with paint, and I'm going to need more paint. And all those construction materials are going to be scaled up and much bigger, and so I'm going to need more wood. Now the wood goes as the volume. The mass of the wood is its density, which is not going to change, times its volume. And the volume is going to go as the length times the width times the height. If this were exactly some kind of rectangular solid, it would be exactly the length times the width times the height. But because we have grooves and possible extra shapes around the back, it's only going to scale that way. So if I double the length, I should double the volume. If I double the width, I should double the volume. If I double the height, I should double the volume. And if I multiply each of them by 100, then I'm going to get 100 cubed times as much, which is a million times as much. Which we basically knew as soon as we realized we were after the volume. Because remember the volume scales the cube of the length. And so if we increase our length by a factor of 100, then we increase our volume by a factor of 100 cubed. Now what about paint? Well paint just covers the surface of things. And the surface of things is an area. And so this area here is a height times a width. This area here is a length times a height and so on. And we know that area scales as the square of the length. And so we know that the area will go as length squared. And so if we increase our length by a factor of 100, then we'll increase our area by a factor of 100 squared, which is 10,000. So in this last question, we have to go from a population density in the country to an average distance between people. Now normally when you say density, you tend to mean a certain amount of something per unit volume. So I might talk about the density of water as being 1,000 kilograms per cubic meter. But that would be a very unusual choice to talk about a population density because then you'd have to worry about how high people were stacked. Normally when you talk about a population density, you're talking about the number of people in the country. And so you tend to talk about the number of people for a given unit of area. So it's people per unit area that describes your population density, not people per unit volume. Alright, so if we have a certain number of people per unit area, how does the average distance between people scale? Well we already know that the area scales as L squared. What that means of course is that the length scales as the square root of the area. So if you take the square root of both sides, there will be some pre-factors in here that you have to take the square root of as well. But the length will be proportional to the square root of the area times the square root of those pre-factors. So then we realize that if the area has gone down by a factor of 10 to the minus 6, then the length must have gone down by the square root of that. And since the square root of 10 to the minus 6 is 10 to the minus 3, what that means is that people are a thousand times further apart in the country than they are in the city. I figured out which way around to put that just from common sense. Obviously if the density goes down, then the distance between people on average is going to go up. Now just to remind you, all of these things we could answer very quickly. And the way you do it is you spot whether you need to convert between a length in an area, a length in a volume, or a volume in an area. So in the first case we saw that the amount of ink was basically proportional to the amount of area. So if we double the point size we have to quadruple the ink. In the second case we saw that the paint was a matter of area, but the wood was a matter of volume. And so we knew we needed 10,000 times as much paint, but a million times more wood. And for the population density we had to spot the population density was something per unit area, and so that the length was going to go down as the square root of that. And once you get somewhat comfortable with these kinds of scaling arguments, it becomes much easier to estimate things that are outside your experience by relating them to things that are in your experience.