That's one reason, then, why I went thru that model just there. Look at the spherical triangle you see. This same spherical triangle in the picture on your screen is now, those are 90 degree corners. Next picture of that. I can't seem to be able to get that series. There are three three such triangles in a series and they should now there is the same triangle with the 120 degree corners it's in the northern hemisphere; and the other one is in the southern hemisphere. And then one more the middle one where there is the equator. Because at the equator they all are 180 degrees. In other words, the angles can get up to 180 degrees. So, what we find then is, the larger the spherical triangle, the larger the sum of the angles. As it works towards I can make a just take the equator there and that's 180 degrees, we've got a triangle that is 180 at each corner.

Now, I bring those same three rods, then, up just a little here and the angles would be, say, 160 at each corner. Moving it up a little further and they would be l50 degrees at each corner, and they would, finally as the triangles get smaller and smaller they will approach being 60 degrees at each corner; but they are always going to be a little more.

The fact that we discover that what you and I were brought up on as a plane triangle as being normal, is a most extreme case of the most local tiny little triangle. That is very important for you to remember now. As we were taught at school, a triangle is an area bound by a closed line of three edges and three angles. A square is an area bound by a closed line of four edges and four angles, all equal, etc. All the geometries that we learned about were areas bound by closed lines. That's the way it was given to us. So all that we accredit about a triangle is what you see on one side of the line it's the little area inside here. Now, the fact is, I said you have to draw a triangle on something, because I'm going to be operational remember. There is no, even if I say an imaginary triangle, I'm going to be imagining one I scratched in the ground. I continually will imagine a special case. I talked with Sonny Applewhite a whole lot about this last night. While human beings are able to discover the mathematics of the generalized case, and though we are able to use the principle, we always have to use it, as I showed you the other day, objectively, in special case. And even though we understand the principle, when we are imagining it and I gave you conceptionality independent of size we will always however, when we make it conceptual, when we make the dots of something, we make the points out of something. We tend to very quickly associate. We'll make it pink or something like that in a blue background if we're abstracting it. But you'll find that the brain has to use special case. Brain is designed for special case. And only mind has it. So the mind can say to brain, "Think about conceptual triangle", but brain will immediately make it special case.

These are great nuances of exploration, but they are all coming out of operational procedure sticking strictly to it. So, when I say to a child, "draw me a triangle", he says "where?" And I say, "Draw me a triangle" So I say, "How about the ground?" So he draws it on the ground. And I say "You've drawn four triangles." And he says "No, I've just drawn one triangle." And I have to prove to him that he has drawn four triangles. So, we're in Philadelphia here, and so he's drawing on the ground here, a little tiny triangle. I say, "When you drew the triangle on there, you divided the surface of the you did it on the surface of the earth, and you divide the surface of the system that you did it on into two areas. You will agree that if I make a circle around the equator that I divided it into the northern and southern hemisphere, don't you?" "If I make the circle a little further north of the equator, I'll have divided the earth into two areas, a large southern and a small northern. If I get a little further north, it's a smaller northern and larger southern." So the little boy has drawn a triangle here, but it has divided the whole earth into two areas. And the, both areas are bound by a closed line of three edges and three angles. So I say, "You have drawn a very, very big triangle of all the rest of the earth here, and it's corners are you think you've got 60 it's corners, then are sixty from they are 300 degrees each. So the big triangle is 300 "He says "I'm not used to a triangle of 300 degrees." and I said "Well, because your school made you so specialized and so absolutely myopic, as not to pay attention to your environment. That's really we've got to really think of the reality, and the point is you have deliberately done something to our earth you have divided it into two areas." And he said "I didn't mean to be doing it." And I said "You thought up to now that you were not responsible, and now you are responsible, you're doing that whole earth." So he said, "Alright, you can give me two triangles a very big one, and a very small one. Where are the other two?" And I said "Concave and convex are not the same." And he can prove that by the reflection of light the diffusing of light on one side, and the concentrating of light on the other and so there is always there going to be a big concave and a little concave, and a big convex and a little convex. You've got four triangles, and you're always going to have four triangles." It's going to be our friend the tetrahedron. The accountability is there. This is a generalization of the tetrahedron as the minimum system in Universe the minimum structure. And it can appear as that kind of a "fourness". They will always be there. There is nothing you can do without it being there. So you can say, "I can hide away." "No, the Universe won't let you do this. Just thinking," I said, divides the Universe into an insideness and an outsideness you didn't mean to do that but you are. You are immediately dividing up the Universe. What right have you to divide up the Universe? Well, you were given this very special kind of capability of the mind. And so you can play with total Universe, and this gets to be quite exciting to feel this spherical triangles and understand that."

Incidentally, if you do any of the mathematics of plane trigonometry, are exactly the same for the spherical just simply because plane triangle is just an extremely limited case of spherical triangle. So the mathematics of the spherical triangle, really, and there is no such thing as plane trigonometry, there is only spherical. And it's dealing with total systems and the beautiful complementations of total systems.

Now, so we say all you have to do is learn the spherical and the plane is included. Give you the plane, and the spherical is not included. Again the advantage of starting to work from the whole to the particular.

Now, something else I was brought up with that schooling. I was taught fractions, and the teacher taught me that I could not have on top of the fraction, elephants, and peas below. You had to have elephants both top and bottom in a fraction. You could not fractionate dissimilar phenomena. That all felt fine, seemed logical, until we came to trigonometry, and they suddenly began to give you sign and cosine, so I said "What are those words?" "Show it to me." So they said "I can't really show it to you because it is a ratio between two it's a ratio between this edge of the triangle and this angle." So, a ratio is a fraction. So suddenly they were giving me elephants and peas and saying it was logical. One reason trigonometry has been difficult to people is because they insist on trying to equate, seemingly, dissimilar phenomena.

But, if we get into spherical trigonometry we have no trouble at all, because we then realize that the edges of the triangle are simply the arc of the central angle of the sphere. So you have central angles and surface angles ALL ANGLES. So your fractions are entirely between angles. That comes in all simple and nice, and gets to feel pretty good. In other words, the way that trigonometry is taught, you absolutely, automatically, cut the kids feelings right out. You say, this is something not like these are signs and cosines. They are exempt from the elephants and peas. This is when they said "Mathematics is something purely abstract forget about all those models." These are the disconnects that I talked to you about when I was trying to find, how do we get back to the conceptual and to our experience so that humanity can understand all of science? And they can!

The more you play with what I'm talking about, the more fun you're going to begin to have, and you're going to find it very easy to take ping pong balls and begin to try out great circles on them, and they're nice to write on ping pong balls, and the colored pens write on them nicely, and it's very easy to get make great circle rulers, so you find out what the diameter of the ping pong ball is, and then you get a little, like a napkin ring, that's just half of that, just the radius in depth, and you sit the little ball in it. And then you just draw on it all these great circles. So you can take the ball it doesn't make any difference, just get any two points and then just connect those points into a lovely great circle. You're going to find it a great deal of fun to play with great circles and have concentric triangles and see the way in which the angles begin to decrease.

The little man, then trying to start with a flat earth, and squares and cubes he said were just great. And just looking at the inside of the triangle inside of the square, looking at what nothing what we do there is teaching him to be absolutely biased. My side is right. My town is inside the wall here. This is, incredibly unbalancing to the little child to be exposed to such bias. I hope you feel more and more with me the sense of responsibility to the child. That little child starting out here, and how easy it was to give them misinformation. How easy it was for parents, just loving their kids to pieces, to say, well that rich man got this tutor, and he must know; so we'll get that tutor, and the tutor tells it his way, and it may be very ignorant. And how quickly conditioned reflexes develop about who is the authority about what.

But the minute you begin to do your own thinking and go back to the experiential basis of things, you can't get fooled. And you continually get better information. It's just so exciting the lovely, clean things you really, suddenly, every time you get understanding instead of something you memorize, some little local thing you memorize isolated from other things.

At any rate I, all my early the globes used to be always fastened to things. I always kept cutting them lose, and finally because you can always set a globe into a circle. As long as the circle is a lesser circle, it will always feel comfortable there. So it can sit on any bowl, any dish or so forth. so you can have your globes and really get feeling your whole earth. I suggest to all of you that you have plenty of globes around and get to seeing things this way.