I made a diagram last summer of the trigonometric functions, and I thought I would complete it for you, and in trying to get it run to your head, because I am a little slower at looking at my paper and putting it on the board. But when I finish it, I think it is going to be useful to you. Remember then that we are always starting with Universe and then we're subdividing, so we start with a sphere our geometry begins with a sphere, or at least with a system, and a sphere is simply a high frequency omni-structural system. The sphere is a high frequency structural system all triangulated. And so it would be approximately unit radius, high frequency structural system. And I'm going to go through some of the arguments of the geometries of the early days, about how you play the game of geometry of Egypt and Greece which is really worth our feeling here, because they took for granted a flat earth, so they started on a plane. But if I'm going to play the game, in starting with totality, I've got to say, "What do I have to prove things, and so forth," and I think you're going to find that it comes out really very satisfactorily.

So, I'll complete my picture here, this is then a section through a sphere. Radius and we just make a working assumption radius is one. It is sub-frequency, it is just unity. Radius is unity. And I have the angle that is considered is called theta, that's from here to here, and this is half theta, "A" is half theta. And it gives, when it's you can have a right to have a perpendicular bisector of the chord, so we can assume it is a 90 degree angle so it is a right angle out of which we then come into trigonometric functioning.

(I have spoken publicly for years and years and years. I have never drank water at speaking till I came here, and always, I was a long-distance runner, and I found that nature makes her own juices, and has her second winds and things, and I know that you don't drink water when you're running you let nature do that, so that I'm a little puzzled by this I think it's a tale end of a cold, I had a very bad cold in late December, and came back from Europe I'd been speaking in Europe and I came back from Europe to speak here in Philadelphia, and I literally couldn't speak it was the first time it ever happened to me. And, so this is some kind of tale end of something going on there, I don't know what it is.. I'm sorry about it.)

Now, coming back to our diagram. I want to complete it, and it will just take me a little bit to do so. Now I want to make a green line here, and I want to have a (there are so many things to keep in your fingers!) and an orange line here, vertically, and I'm going to have a red line this line DB, E up here, and I'm going to make a diagram of that here. I'm just sort of making a side picture, where it's in the diagram on the side that seemed to be useful...(Bucky is drawing this whole time). Now, what's missing here, there's a M missing here. I hope that's clear, this goes there. Now, I think everything is on here, and we are talking about a trigonometric angle and this is angle theta. So she is always isosceles, and then to get the advantage of a right triangle, and all the laws of right triangles, we then have this perpendicular to the chord, and we get another half theta, which is A. And that is the angle we always do our checking at this is what we're checking up on all the time.

And, so, I'm now going to mark what these are. This orange one over here is FD F and goes down to D here. So FD is cotangent. It actually says all these things right on the diagram, but I thought I'd separate them out too. F and O always the center of the sphere. This is F and O and it is your cosecant. And the red one here has the E and O for me here, and this is your secant. And then, E to B is tangent, and M to K is the famous sine s-i-n-e. And this is your cosine. Those two sine sides of the right triangle, the sine and cosine. And so they are of this famous angle half theta, A, the one we're looking at. O.K.?

I think this makes things look fairly simple for you, in looking up you know what you are really finding out. The data you get will be either expressed in degrees and minutes, it could be in sometimes they do it in hundredths of a degree and it can even be done in radians and so forth. But it doesn't make any difference which language you use, you are multiplying so your formulas will be multiplying and dividing one of these things times another all the way through to find out various trigonometric phenomena. I think this is quite, quite a simple diagram, and it was not given to me when I was young, so I really worked on something to have it be seemed to me pretty easy.

Now, I'd like you to think about the game of geometry as played by the Greeks and the Egyptians. They played the game with straightedge dividers. So with your dividers you can strike circle, or you can measure distances straight edge and dividers. You had to demonstrate an unfamiliar geometrical form consists of demonstrated forms and terms of your original constructs. But where as they played the game of making using a straight edge and then taking any point on the line, put the dividers on there and they can strike a circle. And it will cross the straight line at two points, and those will be the radii and they know they are equal to one another, and they are half one diameter. This is really a very simple game.

They could then, from the where the circle crossed the straight line they could go out there with their dividers to the end, put their dividers on the crossing point and strike break into the circle from either end, and they never could prove that the space between was the radius. Therefore, they never assumed the equilateral triangle. And starting from a plane you get into that trouble. And I don't get into that trouble. I'm sure that's one reason why the 60 degreeness has not gotten into the popular game of geometry. A great deal seems to be self-evident, but that's all. And you knew that the dividers seemed to span exactly, but you couldn't prove it.

So , I'm now going to start and, I am going to say I am working on a plane, I am going to make a sphere where I have my dividers and I have a point in Universe and I have a radius going in all directions. O.K. So I now have a surface to work on and I know all of that surface is equidistant from a given point. And I'm going to take any point on the surface of the sphere, and I'm going to strike a circle on the surface of the sphere. You and I call it a lesser circle, but we know that every point on that circle is equidistant from that point on the outside of the sphere. We also know that every point on that circle is equidistant from the center of the sphere. That's quite nice, so now we have quite a little of two kinds of things we have a radius of the sphere that is known and we have the radius to the surface circle which is known. They are both the same divider.

I can now, then, take any point on the surface circle dividers, and strike another point on that same circle can you see that? Now, I'm going to go to more or less the opposite side of the circle. It doesn't make any difference where I come on there some other part of the lesser circle and I find quite quickly with my dividers that the lesser circle really seems to consist of quite a number of increments of this, so I am going any opposite side where it doesn't make any difference, I just take another point on the lesser circle, on the surface, put my dividers on that and strike against the lesser circle again, I get two increments there just look at the globe of the world and I've got a circle up here, but this one here, we'll say, which is about the same radius of it, and I come to an arbitrary point out here, and I strike on this circle here. I now know what that distance is. Quite clearly if I try stepping my dividers around, it takes quite a few divisions, so all I have to do is to go to the opposite side to pick any point I want, somewhere in here, divide it and strike another. I now get two arc sections of that lesser circle, that I know are going to have a unit radius distance now, which is the original radius of a sphere, has the radius of the circle around the surface, and I've got two increments here which are exactly the same. I now am going to run a line back to the center surface point here and two lines from there. This now gives me two triangles on the surface that I know the radii of the edges of those triangles are all equilateral. I know I've got two equilateral triangles with a common apex at the center of the surface circle, which goes down to the center of the sphere. And we know they are all equidistant. So then I was able to take those, and you may remember the way I made the four great circles.

It looks kind of familiar, because I want you to look for instance, here is the sphere line. I got on the opposite point here, and I struck a circle, now I've got these two units here, and all I've really got to do with them is remember how I made them what I have is this and these are all radii, all radii, everything is radii in here that's the reason I constructed it. So there is nothing to stop me rotating around this radii and coming down like that. Now, I have a, I have opened up my I know all those edges are identical in length. And they are all freely rotatable about this, so I now have a hexagon. These are all chorded. I am able to extract the hexagon, then from the sphere, and I really know what it is. Is that comfortable to all of you here? Remember how I did, I first used my dividers starting at this point in developing a sphere, omnidirectional from a point. And then from that I go to the surface of it. And I know every point on the surface is equidistant so I take a point on the surface and I struck this circle out here a surface circle. Then I took any point on that surface circle, and struck over here and got that length. So I got a chord, this is a chord, this we know, we struck that I know all those three points are equidistant from one another now, because they are all constructed that way, by my dividers, the same thing over here. And they don't have to be, you know, diametrically opposite, please understand; because you've proven with your dividers going around that this circle broke up into much more than two, it broke up apparently into six. So I just take, really quite randomly, because this is a hinge, and these are able to hinge exactly like that, so I now have constructed two tetrahedra, edge to edge. I'm able to open the two tetrahedra out in the flat, and now I find them actually consisting then I know I have a hexagon, which every part has been constructed by the dividers, which is a privilege of starting with the whole and working to the particular that you couldn't get to this is synergy the behavior of that whole. It did not permit it if you tried to start with the part. Is there anyone here who can find any geometrical fault with what I am saying? Janet? You're thinking good and hard. Does it seem alright? Good. I haven't found anybody who has ever found any real fault with it. But I'm now really privileged to think tetrahedra. And you can really understand how from there on I can re-establish these four great circles making these four great circles really a part of my game of geometry. It is a very beautiful thing, that you know that you had the circle, so everything is absolutely superbly proven all the way through. Now I have a very nice kind of a geometrical situation where I can then use the diagonals of these squares and so forth, and I can get my x,y,z coordinates. Everything all everything that you are familiar with can come out of here from now on in those great circlings that I gave you. So all the important coordinates that man has ever used are manifest here and constructed so that if you want to take things out in the flat, you are extremely comfortable about what the increments are.

There was, doing spherical trigonometry, which is dealing in the laws of that right triangle it is a beautiful thing right triangle because you by doing so you have one thing that is known. And that is really great. Then, you found that there was sort of constant variation between the whole, it added up to 180, so that if I make one angle smaller of the non right angle, the other one is going to be larger, and the sum total is always going to be 90 degrees. So you've got two very important things known. You've got a known ninety degrees, and you've got a known sum of the other two add up to 90. And they are going to be varying they are going to co-vary with absolute arithmetical accuracy. So this brought about the development of the trigonometric tables, and as I gave you yesterday, the maximum variation you really can get to is then where you have an isosceles 90 degreeness, and it is an isosceles and the two corners are 45 degrees each. So forty five is the inasmuch as they co-vary, 45, the largest gets to be 45 and the smallest gets to be 45. And inasmuch as you have learned that that is so, all you have to know from now on is up to 45. That's why I gave you yesterday all of the prime numbers. There are some people who spoke to me amongst you about the prime numbers, not quite certain why I carried two to the twelfth power, and three to the eighth power and so forth. I simply find that there is, then, trisecting that goes on. You need threeness and twoness, and as you get into multiplication, you need a whole lot of twoness of the prime number two. To give you great flexibility in complex computations. But you don't have, very often that the prime number 43 comes in. One of that is enough to be sure to accommodate, and if it is in the dividend it will accommodate all but there is quite a number of occurrences of the 2's and the 3's and the 5's and you can understand that quite readily. That's why I have many powers of those numbers, and I did not really know, and I haven't got to the control of it yet, how many you would need very certainly; but I what I did was to keep going through and developing, multiplying powers times themselves and sometimes doubling and tripling to see if I came to any interesting numbers. And if you'll look at your book SYNERGETICS you'll find that ever so often the numbers are very regular, and it suddenly comes out absolutely 9 million or whatever it is. There is something incredibly beautiful simple number. And you say, Nature has come in she's clicked here. She's deliberately made it very simple sublimely simple. So I began to give these names that I call sublimely rememberable comprehensive dividends. I kept exploring and multiplying, I have been doing this for years and years, and every so often she came into phase and I've got a listing of those. I used to do all of these things long hand. There wasn't any computer to do it with.