Last updated 5 months ago

“I remembered only that trigonometry had something to do with relations between sines and cosines. So I began to work out all the relations by drawing triangles, and each one I proved by myself. I also calculated the sine, cosine, and tangent of every five degrees, starting with the sine of five degrees as given, by addition and half-angle formulas that I had worked out.”

“While I was doing all this trigonometry, I didn’t like the symbols for sine, cosine, tangent, and so on. To me, “sin f” looked like s times i times n times f! So I invented another symbol, like a square root sign, that was a sigma with a long arm sticking out of it, and I put the f underneath. For the tangent it was a tau with the top of the tau extended, and for the cosine I made a kind of gamma, but it looked a little bit like the square root sign.”

“I thought my symbols were just as good, if not better, than the regular symbols—it doesn’t make any difference what symbols you use—but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, “What the hell are those?” I realized then that if I’m going to talk to anybody else, I’ll have to use the standard symbols, so I eventually gave up my own symbols.”

Invented his own symbols for himself.

“But the whole problem of discovering what was the matter, and figuring out what you have to do to fix it—that was interesting to me, like a puzzle.”

“Just before I came to the fraternity they had had a big meeting and had made an important compromise. They were going to get together and help each other out. Everyone had to have a grade level of at least such-and-such. If they were sliding behind, the guys who studied all the time would teach them and help them do their work. On the other side, everybody had to go to every dance. If a guy didn’t know how to get a date, the other guys would get him a date. If the guy didn’t know how to dance, they’d teach him to dance. One group was teaching the other how to think, while the other guys were teaching them how to be social.”

“They were all excited by this “discovery”—even though they had already gone through a certain amount of calculus and had already “learned” that the derivative (tangent) of the minimum (lowest point) of any curve is zero (horizontal). They didn’t put two and two together. They didn’t even know what they “knew.”

“I don’t know what’s the matter with people: they don’t learn by understanding; they learn by some other way by rote, or something. Their knowledge is so fragile!”

“I gave him a problem: You blast off in a rocket which has a clock on board, and there’s a clock on the ground. The idea is that you have to be back when the clock on the ground says one hour has passed. Now you want it so that when you come back, your clock is as far ahead as possible. According to Einstein, if you go very high, your clock will go faster, because the higher something is in a gravitational field, the faster its clock goes. But if you try to go too high, since you’ve only got an hour, you have to go so fast to get there that the speed slows your clock down. So you can’t go too high. The question is, exactly what program of speed and height should you make so that you get the maximum time on your clock?”

“This assistant of Einstein worked on it for quite a bit before he realised that the answer is the real motion of matter. If you shoot something up in a normal way, so that the time it takes the shell to go up and come down is an hour, that’s the correct motion. It’s the fundamental principle of Einstein’s gravity—that is, what’s called the “proper time” is at a maximum for the actual curve. But when I put it to him, about a rocket with a clock, he didn’t recognize it. It was just like the guys in mechanical drawing class, but this time it wasn’t dumb freshmen. So this kind of fragility is, in fact, fairly common, even with more learned people.”