It had been learned during W.W.II in the riveting of airplanes, when there was an enormous production of airplanes, rivet holes were of the greatest importance. So parts would be made here, and parts would be made there could you brings parts with holes punched in them and bring them in great complex numbers, and still have them all register and have rivets come in. Some of the production tricks the Germans and the English had developed, the United States inherited a lot of these techniques. With extraordinary kinds of controls of light focusing on metals and so forth and using light sensitive things that punched and so forth. At any rate, they discovered, thinking about what kind of strength differences do you get in the airplane where you have two holes and they are a little off like that and the rivet has to be then small enough so it can go between the two because there is a little slack in it. Got into an enormous amount of testing laboratory testing of the relative strengths arrived at where as you got better registering of the two holes as far as the center holes go and the hole sizing, and they found that the strength went up very, very rapidly as you get to greater and greater accuracy both of positioning and the sizing of the hole and the sizing of the rivet that slid in.

For a very simple reason. If rivets if there were some slop in it, it could be, everything was fastened down good and tight, but a big stress on the thing, if there were a little more slop opening in this particularly direction, suddenly she starts to all the thousands, you get a little slop in that side and it would yield in that direction. This starts a shearing action and things begin to go. So if you don't shearing kind of starts it's very much like having a pole that is balanced, you have a pole balanced in your fingers and there is nothing really to keep that no effort to keep that can you keep on that dead center. And if you don't as you get off that, it takes much more effort to withstand, and keep where you like it. So, when I did the Ford Motor Company dome, I also found the best people in the Ford Motor Company, by far, were the tool men. The men who made the mass production tools they are the brownies of Detroit. No matter how the management they are the ones who get things tooled up and get things ready for three years from today when they start selling things. But they are the people who really are keen. And the tooling men liked what I did. Anyway, I designed the rivet holes for the there were some thousands and thousands of these struts many, many thousands, some fifty thousands or so, and they I designed the rivet hole positionings, and the rivets themselves. We kept the tolerance to l0,000ths of an inch! Now bringing a l0,000th of an inch into a big building where the quarter of an inch had been fine, you can imagine the general contractors when I faced them, they said you can't have any nonsense like this, so I said "I have designed all of the tooling, I have designed all of your logistics, as general contractors, simply, I will handle the whole thing for you. You really have to do nothing but just write out the checks and pay the bills here." So I had to put on a show for the Ford Motor Company in my office I took an office in Detroit, where I gave them the complete logistics, the complete design of every part, how every part was to be manufactured, how they would be organized at the job. How it would be assembled the roof, what workman would do each job and how many there would be. I had set out the absolutely whole thing, and the general contractor said he had never had anything done like that before. This was the kind of work his estimators and engineers had to go to work and do, and he found it all done for him. So he was really very agreeably surprised, and that helped me a lot.

It was an operation that was very difficult because the Ford Motor Company had already been spending a lot of money getting things ready for the 50th. They actually had spent, had invested over 25 million dollars in TV shows and the big show they were putting on. And so they didn't want the work they were already doing being messed up, with something going on over head. So I had to build a bridge across this whole thing and I built I used a high-voltage cross-continent mast below to and at the top of it with this bridge going across, I introduced this hydraulic arm that went upwardly, and revolved around. We mounted the dome they were very scared about the danger of the men working there. So all of the it was an enormous steel arm that went around with the dome on it, and people could reach it from the bridge. They didn't have to go up the scaffolding anywhere, and we were able then to continually revolve the dome going up and they kept adding onto just the bottom of it from the bridge, so nobody went aloft, until it was all up and then we let it down onto the actual parapet of the roof, so all of that had to be calculated very accurately. There would be no stresses in that. And the roof was not very beautifully done, so I had to allow for all kinds of come and go, shimming of the thing.

So at any rate the work that I laid out, there were a certain number then of hole pattern. There were six prime variations, and the Ford tooling men made up Class A steel dies to stamp out the parts and to punch the holes. They were able to pre-punch the holes in the flat, before they got into the bending. But everything was under such tight controls, that I kept this down to a ten thousandth of an inch. Now, the interesting thing is that the dome weighted exactly one half of what it would have weighed if it had been laid out by the best sheet metal workers in the world at the tolerance a sheet metal worker can lay out. By I had absolutely under these invisible tolerances. And the dome was you could have exactly two domes for one the same amount of metal, simply by that tolerance difference. That there could be that when you get that kind of economy inherent in competence you can really understand the reason why I would feel that I'm not putting it upon you to take some of your time to begin to get into the subject of the geometry and how we do good calculations and so forth.

Now I am going to come back to man, then as a navigator going around the world and trusting this great ship and all the people in it, and the enormous commitment to this navigator to get you from here to there, going in unknown seas, and you don't know where the rocks are, and you're sailing at night you can't stop the wind blowing in this direction. It's a very hazardous undertaking. And so Napier in England developed some very beautiful rules in trigonometry to simplify the navigator's problems so that he couldn't make what we call a 180 degree error. Because of these quadrants, these x,y,z quadrants you're going around, it is really very easy to make a plus or minus error, and find yourself going exactly the opposite direction from where you ought to be going.

So he simplified this out, and then developed a game of calculation trigonometrically which you could find in Bowditch's Practical Navigator and which was the Bible of navigation for the United States Navy, for the Naval Academy. And you'll still find it there.

But I'm going to give you these rules, because I found it very useful

All through the years immediately, just before the Ford Motor Company Dome a few years after, I found myself being invited all around the world to architectural schools, whether it would be in Ghana or it would be in India and so forth, I would have a class of students, and they would vary from 18 to possible 30 in number and I would teach them, then, all of this mathematics, and I would teach them, then how we calculate geodesic domes and everything; and we would literally make organize production. I would organize the students into aeronautical production. How you go how you develop prototypes and get into production in aeronautics, using their kind of techniques. And there was purchasing to be done and so forth, so really getting the students immediately hooked up with real life. Not only do you get them hooked up with real life, but they found, time and again we had, at Cornell, the President of Dupont happened to be a friend, he had a student, son, up there at Cornell. He was interested. One reason and another there would be somebody you could get in touch with, and you said we are going we are making an experiment, and we really really there is no use in going through this without using the most advanced materials, and we understand you have a little better clear Plexiglass, or whatever it is, and these men would send you a special airplane of materials. So they found enormous cooperation on the part of industry, and began to get enormous insights.

The students I have organized into teams were purchasing agents, and mathematics had a mathematics department, design engineering, and production engineering, installation engineering all the logistics were worked out everything we did; this is what I call COMPREHENSIVE PARTICIPATORY DESIGN SCIENCE. YOU ARE RESPONSIBLE FOR EVERYTHING. You are also responsible for how it's going to be removed and so forth.

Just one sort of last work reflectively on that Ford Motor Company Dome. When we finished lowering that dome onto the and it really worked, and people climbed over it and it was finished. The head of the Ford Motor Company's Engineering Department said "I not only congratulate you" they had been very quiet with me, but he said "I'm going to dismay you very much to tell you what I'm going to tell you, but he said we were so sure you couldn't do it, that we had paid in advance, we paid the contractors, it was going to have to be done so fast, he required a premium, we paid him in advance, we paid a contractor to remove all your unfinished no-good work so we could get so we could go on with our show. We assumed it wouldn't be done. So we're paying him much more than we are paying you for a very beautiful and successful job." Well, they really felt very conscience stricken about it, and as a consequence the Ford Motor Company went way out of their way to tell the Air Force who were going on my radomes and didn't know, they couldn't understand why they seemed to work, and the engineers couldn't calculate that they would work. The Ford Motor Company gave me a really terrific send off and that's one of the reasons why the geodesic domes really did proliferate so rapidly as a consequence of that project.

Coming back now to the Napier developing a trick for the navigator. He got down to very, very simple devices. Assuming you still had held on somehow to your tables, and could look things up if you couldn't the navigator might know his mathematics well enough to know actually how you arrive at each of these calculations, that is a fairly simple kind of formula, you can get yourself within good, sort of practical range, so you know within a hundred miles possibly in some case down to 10 miles within where you are.

Napier made this diagram, and incidentally, in trigonometry if we are dealing with sphericals, we draw a triangle like this we put a little curvature in it, and you can see quite easily whether you are going to if it really is a 90 degree I meant to draw that for a 90 degree corner, so this is a 90 degree. And this corner is big C, this is big A, this is big B. And the side opposite C is little c, and this is little b and this is little a. That's the standard convention in trigonometry. Then, Napier divided developed a diagram where he had and he had big A here, big B, and so this would be little 'a' here, and this would be little 'b' here, and this little 'c' here. And then he put a complementary, little 'c' modifying those three. This is his basic diagram. If you can remember that diagram, then that's all you really need so that you'd never get into trouble. So then he said, I'm going to make a rather poor poem, but you can probably remember it. He said, the sine of any part will always equal the product of the tangents of the adjacent, or the he used the word cosine, he made a rhyme out of and tangents and adjacents he made a rhyme out of at any rate, = the sine of the product the sine of any part will always equal then the product of the cosines of the opposites or the tangents of the adjacents. If you can remember that statement it is not too difficult, the sine of any part will always equal the products (means multiplying) the cosines of the opposites or the product of the tangents of the adjacents. And then I'll show you what he means by opposites and adjacents.