So, this A you now, this is 90 degrees and you've learned that the one thing you know is that this is 30 degrees. So, you can look up the sine of 30 degrees. Oh, excuse me, I'm going to say what I have known in my maybe I know that that is 30 degrees, o.k.? And I know that little b is 22 degrees and some minutes, whatever it is 30 degrees and 20 minutes I know two parts. So what I know is little b, and I know big A. O.K. I've got to find then something called "opps". Well, they are beside each other. Therefore I have to get something opposite to it, so I can say that the sine of little a, and every time I write little a I'm going to put this c above it, the sine of the little a equals these are opps, the cosine of big A times the cosine of little b. Also putting a little like that. Now I've got my formula written out, and I've got to substitute, where the c's occur, I've got to get that out of the thing. These are put in here so you can't make the 180 degree error. So then you convert the c you revert to, so this instead of being sine has to read cosine; cosine of little a equals cosine of big A times the sine of little b. Just the opposite of cosine. Now I've got a formula that I can look up in my tables what the sine and so forth are, and I've got to multiply them, so if you also then have your in those days we used to use logs, so you had the log of any of those numbers and all you had to do was add them, instead of multiplying them. So trigonometric tables were usually also given with the logs in those days, so you looked up your logs and simply added or subtracted, whatever you had to do. Which was fine. So this gave you then, by this time, so now you have learned what little a is, and you go on to find out what the other parts really very rapidly. This is a fantastic, simple thing I've given you here. That's all there is to it.

Now there are many other formulas in trigonometry I assure you, and there are special kinds of tasks to be done, and there are some really quite fancy formulas for doing. But this is the essence. You can get today, Hewlett Packard has a little computer and that's now out at $350, or you can get a $750 one, but it has all the trigonometric functions and everything on it, and you can do all these problems with this little computer. The $750 one they now have programs, strips that you can put in, and so if it were something you did quite regularly geodesic domes, just put in the five frequency, four frequency alternate, and you'd get your answers just like that wham, wham, whatever it's length of every part is of whatever you are building.

Once you've discovered then what the trigonometry is for the radius is one then I showed you here before you want to build a geodesic dome. And what I need to know is the length of a structural member. So what I really want to know is from here to here, I want to know the chords instead of the arcs. So what I do when I get what my angle is, I know, then, that the chord is two times the sine of one-half theta. So, this is quite different. You can't just automatically multiply let's look up to a large angle you can't do that. Because, I want you to realize the difference between the because these things, this would be the chord of that, and that's not at all that number. So you can't just take it's two times the sine of half theta, and just never kid yourself about that. So, just when I have then I began developing what you call chord factors the phrase had never been used before, geodesic domes I could give you the chord factors for any radius, so all you had to know was your radius, and you could do it in meters, or centimeters or anything you want, but, so, as people began to catch on to what I was doing, then they found they could publish all the geodesic domes of various frequencies and the chord factors and that's all you had to know and you start putting together and there's your dome.

So, I can tell you that when I did the first calculation of a geodesic the, I say there were no electric computers whatsoever. There were no kinds of everything had to be done longhand. You did have your log tables and you did have rather poor tables, I didn't have the Edwards ones. So, it was just very clear to me, coming at things the way I do, you're used to my kind of argument, that an omni-triangulated sphere, and particularly if it were tensegrity that is operative, was simply going to make since tension has no limit. Therefore, a tensegrity dome would have no limits, but all the arched domes had very limited clear spans and St. Peter's was the largest in the world at the time of __Mem -few __days(ms?) 150 feet in diameter, and I looked at we really get into some very large sizes, they no longer were building the domes just with bricks and so forth, and having chains around them where you did Santa Sophia or St. Peter's. But they were doing what they called radial arch. They had great enormous steel beams running from the perimeter all the way up, which makes a very long beam. The slenderness ratio made it very heavy, and then they had a centering steel ring and they were all brought to that centering steel ring. And the weights involved were enormous.

Then they began to learn they could make it a little lighter. And then they began cross triangulating. Well this is what came out of my geodesic triangulation so you begin to find you can make those radials a little lighter by cross-triangulation. At any rate, the first my first calculations, I could see that the thing would probably work, and suddenly, supposing there were those who had thought that before, the whole thing was doing calculations so things would come out accurately. It was not a game you could do be just rough about, or things would really collapse. Errors would accumulate very, very rapidly as you went around the great circle.

So, my wife had a little money at this time, she decided to really help me buy time. We bought time, and I'm sure this is the reason why other people hadn't done it, because it really was going to take time. It took me two years to do the first calculations, really to know it was so. And today, anybody with a computer, I mean you really can run this out in less than half an hour you can knock out a dome. And nobody realizes the enormous advantage that has really moved forward to the man with the calculating capabilities. And the computer's carrying out tables to very, very many places and so forth. So that the accuracies are very, very great today. To me, one of the most interesting challenges here was in the calculating capability. If I hadn't done whole number long-hand work I would not realize the significance in numbers.

If you put it into the computer, you just miss it getting your simple answers. I realized I had really tended to make an exploration about the last moment in history when you would have the opportunity to really find something out, and that I'd really better pay fantastically strict attention all the way through here to the significance of everything as I went along.

This kind of talk I'm having with you tonight, then, is to do with a then what I call design science COMPREHENSIVE ANTICIPATORY DESIGN SCIENCE. You deal with things sum totally and in terms of total resources, everything you know about how the Universe is working, and how and why we have the energies available here. Why there is a biosphere. And how you really then employ the physical resources and the knowledge to the highest advantage for all humanity, and if possible to sustain all humanity for all generations to come. That is your challenge, and you must be responsible for how every way you participate in the transformations of nature, employing those principles, responsible for how the things gets where its going to go, responsible for how it goes while it's working there and how you take it away and get it into recirculation again. You must be responsible for the complete cycle. There is no point where you are not until whatever you produce is now melted up and is being used by somebody else. But as far as your using the original resources you make yourself responsible from beginning to end.

I gave you a way of realizing, yesterday, in big patterns the metals that are occurring around the earth, and therefore there is a half way around the world you go to find them all and they gradually converge until they get to maximum separation apart, and then they get into reassociation in preferred ways, and then when they are they finally have such an advantage for man to really justify such a big operation, you have to make them available to the most people around the world, which means you have to send them half way around the world again to make them available. So that is the size of the operation. And, often, really doing things the right way is much easier than doing them the wrong way.

At any rate, I never find myself shuttering at the size of the problem, and everyone of the projects that I have undertaken, and tomorrow we are going to go over a lot of the projects that I have undertaken, were always undertaken on this kind of a basis, and all the students who have ever worked with me have learned how I feel utterly responsible all the way through to humanity for having tampered at all with all this extraordinary phenomena we learn about.

I assure you this kind of carrying on is a very inspiring matter. It makes you tremendously conscious of everybody and why everybody is doing what they are doing at this particular moment I can understand why they are preoccupied. I can understand the fears of the father about his kids going to be able to eat, or whether he is going to lose his job. I understand all of those things. And so I feel very, very kin to everybody. Not at all annoyed at non-cooperation, but you have to find out,. then, how to get it done, and you keep at it until you suddenly begin to find, there are ways of getting things done.

Now, I've been giving you a few sort of clues I'm always looking for the simplest also. So it's nice to get this thing out of the way. That's the language of words, and they should not feel formidable at all, because it is really a very simple kind of a thing, because that is a tangent line, and you can understand what is really cotangent. You can see all of that. It's really, very, very self-explanatory. The only word sine you see this is the withoutness, the openings. This is really to do with angles, how much the angle is open. You can see that, that's a very nice measure because it's within the central angle, I think that is enough of talking about that diagram and talking about Napier, and