## Philosophy 167: Class 12 - Part 3 - The Early Mathematics: Newton's Work While at Trinity College.

Smith, George E. (George Edwin), 1938-2014-11-25

0:06

We have his college notebook when he was first doing math and work in optics. And I'll pass it around. Nice small thing. Dibner had his copy of Ovid's Metamorphoses, it's about this size, annotated in the margins. Meticulously annotated in the margins. So, he could write very, very small, as he did here.

0:30

This is while he's reading von Schouten and Wallace on math. I pulled out select pages. Remember, von Schouten put out the a hugely expanded version of Descartes' geometry, and that was crucial to Newton's mathematical development. But before he got there, there were two textbooks, one by Otrid on fairly elementary algebra and arithmetic, and then a textbook by von Schouten, and this is what it's saying.

0:59

Geometrical propositions from von Schouten. And it's a series of problems. Section one, two, three, four, five, six. It goes up to section ten where he's doing problems one after another. And that is my picture of what he was like. He loved doing problems. He did problems everywhere you could imagine, and he would be a totally satisfied human being, working several hours on a problem with no human contact with anybody else.

1:33

When he heard about the problem of King Solomon's temple, and the description of it in the Bible, the architecture. And people couldn't figure out how it's structured. He goes back to his room two or three weeks later emerges this, the document is part of Dibner collection down now at Huntington library.

1:53

His account of King Solomon's temple. He's worked out what the temple had to be like from architecturally, from the description in the Bible, and it's just so typical of him. He has this huge book on prophecy, trying to locate when various things happened in the past astronomically. That's him.

2:11

He's gonna figure out where everything happened in the past. He is a problem solver. And a problem-solver with paramount ability at problem-solving, not an abstract theoretician first and foremost, but a problem-solver. And that's why I included these, just so you can see a slight sample of his problems.

2:30

This is a diagram by Tom Whiteside of how Newton got to the calculus. Which he dates, Whiteside dates it as October 1666, and he starts it in 1664. So we're talking here, essentially, two years. The main thing about this, you can read it on your own. The main thing about this is telling you how totally eclectic Newton was.

2:58

He would see one person do a problem. He would see another person do a different problem. He would start putting the two together. He had just no reluctance to do that. And he was very, very good at doing it. The famous statement he made, correctly attributed it him, it's in a letter to Hook.

3:16

About standing on the shoulder of giants. Hook had asked him how he had come to his new mathematical method, and his comment had nothing to do with the Principia physics at all. That he had been standing on the shoulders of giants, and I'm sure what he had in mind, he had John Wallis, he had von Schouten, he had Hutty writing in von Schouten's Cartesian geometry.

3:40

He had, who else? Barrow himself lecturing at Cambridge. I think those are the principal people up here, all of whom were moving somewhat close to working out how to do tangents and other problems. Wallace had developed, the binomial series and realized he had done some infinite series. He had also begun using what's called the method of indivisible which is infinite decimals.

4:11

All of these lines come together when Newton. First place he picks it up, 1665 he realizes what we now call differentiation and integration are the compliments of one another. That's called the fundamental theorem of calculus. That's the one Lydon has published in 1682 with his first publication on the calculus.

4:35

That's a long time after Newton as you can figure out. A couple comments, I am going to go through a series of pieces on mathematics on this, Tom, excuse me on Whiteside, Whiteside was somebody I knew very well. Very hard to call Tom Whiteside a personal friend because of his.

4:56

He had a temper, he actually spent a night in jail for punching somebody who gave a lecture on the history of mathematics and he got mad enough he walked up in front of the room and hit him. That's the personality, that's totally him, okay, and so I'm slightly reluctant to call him a friend because I'm afraid he'll come down from heaven and punch me in the nose.

5:18

But we ended up being very close late in his life because of our mutual tide burner. So I had trouble not calling him Tom. It's like Bernard Cohen, both of them, I'm too close to. But in spite of what he says here about dot notation, the dot notation Newton is famous for, he didn't come upon until after the Pricipia in the early, late 1680s, early 1690, he did not have perspicuous notations for his calculus.

5:48

Our notation is almost entirely our notes derived notation. Newton's picture is derived from Barrow. Namely, take a point and have it moving through space. It describes what we would now call a function, a term he never used. And it moves at a certain rate. That he called a fluxion.

6:16

It's its derivative with respect to time. And if you have the fluxion, the derivatives with respect to time, the function itself is called the fluent. So fluent have fluxions. You'll read about it in a moment. A fluent is what we would call a function and the fluxions are the differential elements versus time, of this moving dot.

6:38

So, that's the way he started thinking about the whole area that became the calculus. Everything was done with respect to time and this moving dot. Meanwhile, and just as important, he realized something that Wallace didn't realize. If you expand a binomial series and don't use integral, integral values of the exponent.

7:02

The binomial series you know what it is. Square of a and b is a squared plus 2ab plus b squared. Cube goes one three, three, one. And that keeps expanding. Newton had the cleverness to move away from integers in generate infinite series and realize he could draw all sorts of conclusions in the limit from those infinite series.

7:27

That's the thing up there about the binomial series in the winter of 1664. That, together with the picture of the moving dock, is the cornerstone of how he came upon almost every thing he did in math initially. And neither of those were completely original with him, though in both cases, he carried them much, much further than Barrow had, or Wallace had.