 So this video will be about the question of what Galileo thought the shape of a hanging chain was. So many sources claim that Galileo thought a hanging chain was a parabola. The actual shape of a hanging chain is something called a catenary more or less by definition because catena is Latin for a chain and catenary is pretty much by definition the shape a hanging chain makes. However, it's not quite a parabola. The problem with the statement that Galileo thought a hanging chain formed a parabola is that it's obviously stupid. What we should do is remember that Galileo was one of the greatest of experimental physicists. He pretty much founded the entire subject of experimental physics and before making statements about the form of a hanging chain he would have actually checked. So he would have got a chain like this and picked it up and you can see it sure does look like a parabola. In fact, it's really close to a parabola and it's quite hard to tell the difference between a hanging chain and a parabola. However, if you bring the ends of the chain in it actually becomes less and less like a parabola and if it stopped wobbling and you checked it against the parabola you would see it is definitely not a parabola. So we should take a look at what a catenary and a parabola are and see what Galileo actually said about them. So here I'll just bring up my camera. So let's just recall what a parabola and a catenary is. So a catenary kind of looks a bit like this and has the equation something like y is the hyperbolic cosine of x which is e to the x plus e to the minus x over 2. A parabola on the other hand looks something like this and will have equation y equals 1 plus x squared over 2. And these are very similar because if you expand the catenary as a power series you see this is 1 plus x squared over 2 plus x to the 4 over 24 so on. So for small values of x these are really both very very similar. Now let's see what Galileo actually said about them. So this is a copy of his book The Two New Sciences and he's discussing catenaries up here. So he said over these two nails hang a light chain of such a length such that something or other the chain will assume the form of a parabola. So this this form be marked by points in the wall we should have described a complete parabola etc. Okay so this is a bit of a problem apparently or I was wrong and all the people who said Galileo was mistaken were right after all. In fact the translator has even added a little helpful footnote here where he points out that it is now known that this curve is not a parabola but a catenary. So that seems to settle that well obviously it doesn't obviously I'm not wrong because I wouldn't be making this video if I've been wrong. So if we look ahead a bit we find another section of the same book where he says here I must tell you something which will both surprise and please you namely that a chord stretch more or less tightly assumes a curve which closely approximates the parabola so he doesn't say they're equal he says they're very similar and he says if he goes on you'll observe that on slackening the chain more or less it bends and fits itself to the parabola and the coincidence is more exact in proportion as the parabola is drawn with less curvature also to speak more stretch so that by using parabolas described with elevations less than 45 degrees the chain fits the parabola almost perfectly so what Galileo is saying is that provided this angle here is at most 45 degrees the chain fits the parabola almost perfectly and he quite clearly implies that they are not the same if you go further than that. So what's going on with this earlier statement where he said it was a parabola well if you look carefully what he's doing is he's describing a method for drawing approximate parabolas and the method he gives for drawing approximate parabolas is extremely accurate in fact the error is probably less than the thickness or whatever pencil you use for drawing so okay when he says the curve you get here is a parabola what he means is it's approximately a parabola within the error of the thickness of whatever pencil you're using it's kind of like if you go around saying the earth is a sphere well that's wrong the earth isn't quite a sphere because it's slightly squashed because the earth is spinning but it's not really wrong because it's pretty darn close to being a sphere and when Galileo says a hanging chain is a parabola in book two what he means is it's really close to a parabola and he knows perfectly well that it's not exactly a parabola so i guess the point of this is if you hear that someone really smart like Galileo said something that is easily checked to be false you should be rather skeptical and maybe they didn't actually say that after all