Let’s look at the Euclid of Alexandria, the father of geometry and his contributions in celestial mechanics and orbital calculations.
Show Notes
- Who Was Euclid of Alexandria?
- Importance of Euclid’s work:
- Euclid’s Contributions to Mathematics
- Influence on Astronomy
- Euclid Legacy
Transcript
Human transcription provided by GMR Transcription
Fraser Cain [00:00:49] Astronomy Cast Episode 733 Euclid of Alexandria. Welcome to Astronomy Cast, our weekly fact based journey to the Cosmos to help you understand not only what we know about how we know what we know. I’m Frasier Keen. I’m the publisher of Universe Today. With me, as always, is Dr. Pamela Gay, a senior scientist for the Planetary Science Institute and the director of Cosmic Quest, Hip hop. How you doing?
Dr. Pamela Gay [00:01:09] I am doing well. It is raining and raining and raining, but it really looks like Halloween is supposed to look. Even though we’re now a week past Halloween. I’m just going to embrace the fall.
Fraser Cain [00:01:27] Yeah. Yeah. I mean, always. I’m always stunned by how beautiful the fall is here on Vancouver Island. The trees have turned various shades of of red and yellow and orange. And it’s just in the collective. They fall down and and move around like snow drifts of of color. It’s amazing. I love fall here on the island.
Dr. Pamela Gay [00:01:51] And you also get the salmon, don’t you?
Fraser Cain [00:01:54] Yeah. That occurs. Carnage at a whole other scale where you have just thousands of salmon swimming upstream through all of the rivers and streams, and then they die flop around on the bank. Then you get tons of wildlife, bears, cougars. One of the cougars, like the many white bears. Tons of seagulls, Eagles eating ospreys, various birds and and mammals feasting on the banks of the of the rivers. The smell is pretty nasty, but it is, you know, it’s just a sight to behold. Seeing this level of of a sort of animal migration happening. And we end this is the one of them. The other one that we get is the hearing in March. And that’s even more spectacular because it’s the largest biomass migration on planet Earth.
Dr. Pamela Gay [00:02:46] And these are just like little fish, aren’t there?
Fraser Cain [00:02:49] I mean, they’re like maybe ten meters long, like, you know, half a foot. Yeah, but there is just like a trillion of them. Yeah. The exact number. But it is like and they all shop around our area. So, so we happen to be at the just the epicenter of the largest biomass migration on planet Earth. And it is just it’s crazy. I mean, the herring change the color of the water so that it looks like like that sort of, you know, the way you’re in like the beaches in, I don’t know, the Caribbean, that sort of. Yeah. Blue color and that’s what our oceans normally very dark, kind of dark blue, dark, almost green blue right pouring on black in the winter time suddenly turn these sort of and it’s just like it’s the it’s the fish getting it on. And then the the beaches are covered in the eggs of the like like again ten centimeters thick. You could you, you have to wear boots to walk through the eggs on the beach. It’s the craziest thing.
Dr. Pamela Gay [00:03:58] That seems like such a waste of energy to have so many of the eggs on shore.
Fraser Cain [00:04:03] Well, that’s. No, they they they want that. They attach them to seaweed and rocks and all that kind of stuff. And then when the water goes out, then they’re revealed to the surface. And obviously all of the birds stuff come in and eat them. But then the water comes in and they’re protected again. And so that’s their that keeps them away from the deeper water stops, the bigger fish like the big salmon coming in and eating them all up. But you get just an infinite amount of killer whales and humpback whales and right whales, gray whales and California sea lions and harbor seals and eagles. You know, you can go out and just see, I don’t know, 500 eagles, all just hovering around waiting for their chance.
Dr. Pamela Gay [00:04:45] We get that, too, but for very different reasons. It sounds absolutely amazing.
Fraser Cain [00:04:51] Yeah, yeah, yeah. It’s a it’s a it’s an amazing experience. I mean, we take we take advantage of it every year we go out and and you know, you get, you get an update on your phone where the salmon spawn is happening and then you drive or catch a ferry or, you know, take a boat out to where it’s all happening. And yeah, it’s a pretty amazing experience. Yeah. If you if anyone, you know, people always thinking you have to come here in the summertime.
Dr. Pamela Gay [00:05:14] Now.
Fraser Cain [00:05:15] Because the weather’s really nice. You want to come in the wintertime because the skiing is really good. But no, you want to come here in March to see the hearing migration. All right. Should let’s get on with the show. Last week we talked about the mission. This week we’ll talk about you could of Alexandria, the ancient Greek mathematician who inspired the mission. Let’s learn about his life and the groundbreaking work that made so much of our modern mathematics possible. And we will talk about it a second. But it’s time for a break.
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Fraser Cain [00:06:46] And we’re back. So yeah, in my intro I said, He’s an ancient Greek mathematician, but he comes from Alexandria. So what’s the story? Where where did you could live? And you know, what? To what nation did he hail from?
Dr. Pamela Gay [00:06:59] So this was during the the great Greek empire of Alexander the Great, who founded his namesake city of Alexandria. And so it is thought and very, very little is known about the actual life of Euclid, but it is thought that he was a Greek living in Alexandria under Ptolemy, the first. And it’s actually a story of Ptolemy saying to Euclid, Is there not an easier way to an understanding of geometry than reading all of the elements? And Euclid responding? There is no Kings Highway to geometry that that dates when Euclid was most likely alive as around 300 B.C. most reports on his life were told 100 or more years after him. But he is someone who essentially defined modern mathematics more than 2000 years ago and has has had things named after him over and over because of what he accomplished in 300 B.C..
Fraser Cain [00:08:18] So give us a bit of history. How did he I mean, it’s so tricky, right? Because the records aren’t great. So, you know, what do we know of of his life?
Dr. Pamela Gay [00:08:28] Well, it is thought that he may have attended a school in Athens, that he was a couple generations removed from Plato in terms of academic generations. And he was one of the first mathematicians to start laying out the foundations of how to think about mathematics. And this was really the thing he did that that in many ways was unique. He started out by saying, okay, we’re going to put forward definitions, we’re going to put forward postulations, we are going to then define common notions, and then we’re going to put together all the propositions of what is a line that that thing that we all learned in school of any two points connected is a line. That’s a line segment. Keep going. You have a line. A line is infinite in duration or distance. I guess on extent that that comes from Euclid.
Fraser Cain [00:09:46] Right. And and so, like, where did he specialize? His mathematics?
Dr. Pamela Gay [00:09:54] Well, he’s he’s known as the father of geometry, Euclidean geometry for the longest time when we thought of geometry as a human race, it was with this notion of two parallel lines will always stay the same distance of heart across the entirety of their length. That’s one of the main defining characteristics of Euclidean geometry, and he worked to define all the different things that that we need in order to to understand quality of angles of areas. The idea that if you have two different things that are both equal to a third thing, those two things are equal to each other. That if A equals C and be equal, C equals B, that was Euclid defining it, laying it all out. So so he started with that and his most famous work, the elements extended across 13 different volumes in the days when books were written by hand, transcribed by hand across centuries. So he started with the Pythagorean theorem, the series of angles laying down. This is this is geometry. And then he moved on from there to start talking about concepts that would later go into the foundations of algebra. Now, algebra didn’t actually exist in its modern formulation for quite some time. He predates algebra, which I find deeply amusing. No real reason. I just do. From an. And he went on to take on things that feel extremely modern, like the idea that there are an infinite number of prime numbers. Was one of the things he laid out the ideas of conic sections of of what it means to cut through a cone in different ways and get different kinds of curves. He spent an entire volume looking at platonic solids. This this was fundamentally what he worked on doing, documenting the bejesus out of stuff.
Fraser Cain [00:12:26] Right. Right. All right. We’re going to continue this conversation, but it is time for another break.
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Fraser Cain [00:13:33] And we’re back. So what were some of the sort of big accomplishments then? You know, he had written how many talk in his 19 months, Was that right?
Dr. Pamela Gay [00:13:43] Just in the elements, there was 13 volumes, 13. And that was by far his most lasting work in its original. He also did books strictly on conic sections. And there was a later mathematician that I apol Apollo and honor. I’m going to mispronounce this. I’m so sorry. I’m going to surprise no one in mispronouncing this. A Paul A.S. wrote a book on comics that was built on the foundations of Euclid’s work and supplanted it so that Euclid’s work doesn’t remain in in its full entirety. But. With all of these works. What he was doing was, for the very first time going about creating the mathematical definition of rigor of this is how we define ideas and. The the root of everything he did was basically five postulates and all of math. Since then that has looked at geometry has been built on these five. That’s pretty important. So there’s the line already brought up that if you draw a straight line from any point to point, that was him to produce a finite straight line, continuously in a straight line. So you go through the two points, keep going to describe a circle with any center and distance. So the definition of a circle is you define a point, you define its radius, and you go around. And that is the definition of a circle that if a straight line falling on two straight lines makes the interior angles on the same side, less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which all the angles are less than that of two right angles. So. So that last one is a mouthful, but.
Fraser Cain [00:15:59] This is reminding me of my grade nine math course.
Dr. Pamela Gay [00:16:01] He defined grade nine math.
Fraser Cain [00:16:04] Yes. Yeah, yeah. I mean, I remember you to do this. You would have this. You’d have this sort of weird object with a bunch of lines crossing at different angles and stuff. And then it would be like this process of elimination where you would try to figure out, well, you know, the angle of this thing, you know that those two are opposite and they’re going to be the same angle and the cosine. Yeah. And then those two are, you know, add are all on a straight line and they have to add up 280. And so you subtract one from the other and they get to the number. Right. Like all of that stuff is the Euclidean geometry. But, but also what’s really interesting is that then people figured out non-Euclidean geometry later on that, you know, is almost like that. You can then use the fundamentals to then describe other kinds of shapes that are outside of the more familiar flat surfaces, which is kind of amazing too.
Dr. Pamela Gay [00:16:59] And this is where the the idea of signs and cosines and tangents, all of that would come later. But the basic ideas of number theory of of his common notions that he had to lay out because if you don’t state these things, they don’t exist. So there is the if A equals seem B, c, A equals B, well, then that had to get extended to the if you subtract C from A and you subtract C from B, then the new number is going to be identical if A and B are identical. Right? So these ideas of if you subtract the same thing from two numbers, that from two things that are identical, you get the same thing. These ideas were. Things that shopkeepers used on a daily basis. Yes, but not magically had to be laid out. One of his things that he wrote down was the hole is greater than the part. And I just love the fact that somebody had to write that down.
Fraser Cain [00:18:10] Right.
Dr. Pamela Gay [00:18:11] Yeah. And Euclid wrote it down.
Fraser Cain [00:18:13] We’re going to continue this conversation, but it’s time for another break.
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Fraser Cain [00:19:18] And we’re back. So I think, you know, we want to sort of create a through line from Euclid’s history is a mathematician to the Euclid mission itself and to what was his influence on astronomy?
Dr. Pamela Gay [00:19:32] Well, here I think the thing that is most key to that mission is he defined geometry in which two parallel lines stay parallel. And with the W map mission, it was determined that in our universe, two light beams that shine parallel will always remain parallel. Our universe in its structure, as far as we know, is defined as Euclidean geometry. We’ve talked about this before. One of the postulated geometries of our universe is a hyperthyroid, basically a four dimensional donut in which any two lines going around the surface of this four dimensional surface will always stay equally separate from each other. It’s it’s right. The Simpsons geometry.
Fraser Cain [00:20:27] Right. But a cube will work too. Like two parallel lines going along the outside of a cube will maintain a parallel nature. They’ll have to go around the edges of the cube, but they’ll remain parallel. Or a dodecahedron like you can imagine a whole bunch of different shapes. And then one that is also useful is that idea of a of a Taurus. But the point being that if the universe was a was a sphere or a saddle shape, then parallel lines wouldn’t remain parallel.
Dr. Pamela Gay [00:20:54] Then they would either come together or diverge.
Fraser Cain [00:20:58] Right? So and so that and I know we call the universe flat rate, even though it’s not flat, it’s three dimensions, but it is flat in in three dimensions as this, you know, if that makes any sense. But but and so I guess what you’re saying is that that flatness, that that parallel line that he had defined 2000 years ago plus.
Dr. Pamela Gay [00:21:21] Is our geometry.
Fraser Cain [00:21:23] Is our geometry of our universe. And the commission depends on that to make his measurements of the of the nature of dark matter and dark energy and concentrations and so on.
Dr. Pamela Gay [00:21:35] Yes. And I have to say, I deeply appreciate the idea that they also named the mission after someone that we’re not even sure what university he went to. We know so little about this human that we cannot be troubled by him. So we we are in this place where we can commemorate the knowledge, created the history, created the concepts created in a cringe free way. And it’s nice to have heroes that are cringe free occasionally.
Fraser Cain [00:22:14] Right? Well, I mean, we don’t know. Like if someone we don’t.
Dr. Pamela Gay [00:22:17] Know, maybe.
Fraser Cain [00:22:17] Someone will dig up some ancient papyrus that that shows the terrible things that he did. And, you know, as a as a person of that time.
Dr. Pamela Gay [00:22:26] We’re sure they existed.
Fraser Cain [00:22:27] But yeah, who knows. But of course, he worked out of out of Alexandria, probably had a lot of his documents done in the Library of Alexandria which burned. Yeah. And lost an enormous amount of, of human history. So, you know there’s a couple of other parts that I think are really interesting. You know, after Euclid, we have people like Ptolemy now. Not now. You mentioned that he or he lived under the Ptolemy, the first the Ptolemy of the later right.
Dr. Pamela Gay [00:22:57] There was a Ptolemaic dynasty, great leaders out of Egypt. So there was Ptolemy, the astronomer, who would come much later. It is steeply confusing. Yes. Last names, first names. People just had a name and a place and occasionally a number.
Fraser Cain [00:23:18] So yes.
Dr. Pamela Gay [00:23:19] There were many.
Fraser Cain [00:23:20] Ptolemy’s Ptolemy, the astronomer, you know, he wrote this famous book, The Omega, too, that defined the movements of the planets and sort of proposed that the Earth was at the center and you had all the planets going around it. But but the movements of the planets were based on these conical sections that Euclid had defined.
Dr. Pamela Gay [00:23:45] That, yes, that’s entirely true. Although I they were looking primarily at spheres at that point. So with with the all my guest, it was spheres within spheres within spheres. And it would be Kepler that said no. Now whether I like it or not, and he did not like it, they had to be ellipses. And that’s where we realized, we can have things on escape trajectories that are hyperbola. These are. Hyperbola. Rather, we can have parabolic orbits, we can have all these different ideas and and so conic sections. It was an evolution for Kepler. He went from those platonic solids that were so well defined in Euclid’s elements all the way out to the conic sections that, according to a polynomials, were also well-defined by Euclid.
Fraser Cain [00:24:47] Right? Right. And so, I mean, you got Copernicus who placed the sun at the center of the solar system, but he relied on those Ptolemaic and I guess Euclidean spherical slices to describe the motions of the planets. And the math didn’t work out. And you had to go back to Ptolemy to say, no, no, the epicycles, the spheres in spheres. That’s the way. And yet Kepler came around and provided back to the original Euclid Euclidean clinical sections to say, No, no. Euclid provided the answer. We just didn’t see it. You can get both the sun and the center of the solar system and have the planets follow these nine circular paths which are described by slices to a cone.
Dr. Pamela Gay [00:25:39] And if you’re ever forced to write geometric proofs back when you were in school, Euclid is probably one of the people you can blame for that because his way of writing down here, the definitions here are the postulates. Here are the notions. Now let’s propose things and prove them. I’ll get back to Euclid. And one of the interesting things I came across while preparing for this is second only to the Bible is Euclid’s elements in numbers of languages. It has been translated into and additions that have been published and and that in many ways is is thanks to the Arabic peoples who during the dark Ages took this, ran with it with with algebra that was getting defined and brought into the forefront at that point and throughout all the worlds and geometry is relevant.
Fraser Cain [00:26:40] Yeah, yeah, yeah. It’s interesting. I mean, you know, we’ve done a couple of shows talking about the influence of the Arabic astronomers. Most of the names of the of the stars are given to Arabic astronomers.
Dr. Pamela Gay [00:26:52] Yes.
Fraser Cain [00:26:52] And and they relied heavily on Euclidean geometry for measuring the positions of the stars and and using it to create these incredible catalogs at the same time. And so, yeah, as you say, you know, I mean, even to Newton, I mean, when you look at the motions of the of the moon around the earth and the planets around the sun and and, you know, Newton worked out how the gravitational field works to, you know, to keep things going in orbit and so on. He was relying on Euclidean geometry to be able to do all of the math that made all that possible. And so, you know, and again, is this through line, you know, to from from Euclid to Ptolemy to the error of astronomers to Newton to tortured ninth graders to the mission that was named after him.
Dr. Pamela Gay [00:27:49] And we live in a universe with his geometry on a planet that has spherical geometry and do not mix the two. You will only find sadness.
Fraser Cain [00:28:01] Right? Right. Thanks very much.
Dr. Pamela Gay [00:28:03] And thank you, Fraser. And thank you to everyone out there who is part of making this show possible. We can’t thank all of you at the end of these episodes, but I’m going to thank those who donate $10 and up a group of you each week. This week, I’m going to thank Abraham Cattell, Alexis, Arctic Fox, Bart Flaherty, Bob Zawacki, Brian Cagle, Cami Rossi and Cooper David Diane Philippon, Dwight Elk. Evil Monkee. Frank Stewart, Georgie Ivanov, Gordon Dewar’s hot dog. Sure. ROCCA And those letters. I don’t know what to say. Janelle Just. Collins Jim McKeon, Joel Stein, Jordan Turner, Kate and Draco. Kenneth Ryan, Christian Margaret Holt, Les Howard. Mark Schneider. Mathias Hayden, Michael Pitroda. Mike High Z. Paul de Disney. Peter. Robert Handel. Sam Brooks and his mom, Scott Briggs. Simon partnered the bench squish squash time Lodeiro William Andrews. I am thrilled to see so many new names and I am so sorry for what I have just done to them. Thank you. Thank you. Thank you.
Fraser Cain [00:29:22] Thanks everyone. And we will see you next week.
Dr. Pamela Gay [00:29:24] Bye bye. Astronomy Cast is a joint product of universe today and the Planetary Science Institute Astronomy Cast is released under a Creative Commons attribution license. So love it, share it and remix it. But please credit it to our hosts, Fraser Cain and Dr. Pamela Gay. You can get more information on today’s show topic on our website. Astronomy Cars.com. This episode was brought to you. Thanks to our generous patrons, unpatriotic. If you want to help keep the show going. Please consider joining our community a patriarchy slash astronomy cast. Not only do you help us pay our producers a fair wage, you will also get special access to content right in your inbox and invites to online events. We are so grateful to all of you who have joined our Patreon community already. Anyways, keep looking up. This has been astronomy cast.
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