Philosophy 167: Class 12 - Part 4 - Newton's Development of the Calculus: Infinite Equations, A Treatise of the Method of Fluxions, and Curvature.
Smith, George E. (George Edwin), 1938-
This is the first tract on Calculus. It's called Resolving Problems By Motion. And that's what he's doing. He's getting tangents to geometrical lines. He's working out what we would call derivatives. You can even see down here that that's what he's doing. I'm not gonna dwell on this. As I say, it's the first one.
It's got calculus relatively laid out. It's very much working with symbols. He has the diagram there, but everything else is done symbolically, something he abandoned later in life when he cease trusting symbolic methods. But this is the first tract and this did not excuse me, this did not get circulated.
Barrow clearly saw it and that's when Barrow realized he had a student on his hands who was fairly exceptional. In fact you can even make the argument, Barrow gave up his position and vocation of professor of mathematics, and went to theology. Because he decided that Newton was so far superior to him.
Newton. He couldn't match Newton mathematically. This is. This is from Tom Whiteside. This is volume three. Volume two, I'm sorry this is the tract Newton wrote. It's a fairly thorough tract in calculus after Mercator had published his book on rhythms and at that juncture Newton was quite worried.
I've given you this as a handout because Tom gives in the footnote the story of this text. The little bit at the top doesn't tell you much. Do notice the title though. On the analysis of infinite equations, an infinite equation is an infinite series that's equal to something, and you've gotta solve for the unknown on the other side.
Or the other way around. The unknown can be the quantity that the series sums to, or it can be the quantity that is being expanded in the series. You can read for yourself the history of this text, including this stuff about Leibniz in here and what Leibniz got out of it.
We're gonna worry about the Leibniz Newton controversy after we read the Principia, because it starts in 1710, and it's way out of time now. The more interesting text is this one. In 1671 Newton decides to write a comprehensive treatus, called a treatus, on the method of fluxions, that's derivatives and infinite series with it's application to geometry of curved lines.
It runs about 190 pages. It was, well, I'll pass around the English version of it. It fills much of this book. The English version was published in the 1730s and used a late notation. The original does not look quite like this. But let's just start with see and these are in effect chapters.
So let's just run through them. On the solution of equations by infinite series, that's the introduction. Followed by from flowing quantities given to find there fluctuations. That's to find their time derivatives. From the given fluxions to find the flow in quantities. That's to integrate to get the function.
To determine the maxima and minima of quantities. To determine tangents to curves. To find the quantity of curvature in any curve. I'll come back to that one. To find the quality of curvature in any curve. To find any number of curves that may be squared. That's any number of curves for which you can get the area underneath them.
That's integration. To find any number of curves whose areas may be compared with the conic sections. To find the quadrature of any curve proposed. To find any number of curves that may be rectified. Rectified, is the length of a curve if it's laid out flat. The one you all know, that everybody knew forever, is forever, is the length of the circumference of a circle is pie times its diameter.
Now, you want that for all sorts of other curves. And he's giving you a general method for doing it, for any cur that you can express as an infinite series. And it's our way. All these things are our way, there's not real difference other than he's doing things with respect to but I'll assure you in a moment they're the exact formulas that we now use.
Define any number of curves whose lines may be compared with any curve assigned and to determine the length of curves. That's, as you see, it's in English in the 1730's. It runs somewhere around 190, 195 pages. He wrote this whole thing. It's a quite finished text. He tried to get a publisher, nobody would publish it.
Had it been published, we would never had heard of Liedness in conjunction with the calculus. Because Liedness hadn't even started doing math in 1671. This covers a century of full first year college course in the calculus. Through and through when the original lousy notation and unfortunately in the English version of notation.
I'm going to show you one example from this. The book is amazing. He's just got all of what is. I learned calculus from volume I of Korantz differential integral calculus. And there's nothing in that book other than the problems, which are post Newton, that is done here. Volume two is different but it's getting ahead.
So this is on curvature. He introduces the idea by taking two Perpendiculars off of a curve, having then intersect, then move the two points closer and closer until they converge on a point. Geometrically, that's gonna be the radius of curvature at the point. That's also the evolute line from Huygens, but this is done before Huygens had published anything on it so.
well ahead of time and then he goes over here and now I repeat the dot notation which means differentiation with respect of time. That was introduced later but its in the english version. Z becomes a derivative, derivative of Y with respect of time divided by the derivative effects with respect of time, is what we call DYDX He goes through this, does the derivation and gives you our formula for the radius Foley laid out in depth.
The other thing here is notice the little zero and how he goes from this expression to that one, where the terms involving the little zero drop out. That's his notation invented to represent limiting quantities that disappear. And one of the reasons he abandons symbolic mathematics is he gets very uncomfortable with the idea we divide by little o in one step and then we cancel the term Treating little o as zero, and the next step, that's inconsistent, you can't divide by zero, but you divide by zero then do it, the conclusion is you can't give a rigorous specification of limits within this frame work, so he gives one geometrically you'll see it when we get to the the first section of the lays out.
A geometric way of doing the calculus. Neither here nor there. I emphasize the fact that he's totally adept at symbolic methods in 1671. A little over 10 years later, he's gotten truly beyond comfortable with whether they're rigorous. Well we don't get rigor till we get to vier Strauss and Contour in the 19th century.
It's striking. Newton's definition of the limit which is the first step in the Principia, proof step in the Principia, or so called proof is the dedicant definition of the limit. I mean just is quite literally the dedicant definition. We'll get there beginning of next semester so I'm not gonna worry about it now.
But the other reason he gets uncomfortable with symbolic math is on his view, he has a whole book on this called Arithmetica Universalis. It's a rigorous development of symbolic arithmetic through algebra. It's in this one, the complementary volume. He really does understand the rigor of symbolic methods, but the rigor is in the form of the symbol stand for numbers.
Numbers can't capture geometric magnitudes, because geometric magnitudes can be incommensurate with one another But the relationship between two geometric magnitudes is exact, it just can't be represented by numbers. So there are two things that push him to wanting to do all math involving motion and all math involving limits off of geometry.
One is geometry is exact and symbolic methods are not exact. Unless they're working with rational numbers. Well, that's not quite right. Rational or algebraic numbers. And the other thing that drives him to it is he just doesn't see any way to do limits rigorously when you're working with numbers.
And that of course, was the problem that Vier, Strauss, etc. had to deal with. They had to give a definition of real number that actually permitted limits to being well defined. So Newton wasn't crazy about this. Now, you asked about the symbolic method, and you've heard me say this before.
Leibniz's view of symbolic methods was that they were wonderful for one primary reason. You could math without having to think. And he said that very prominently and Newton found it a total abomination, the very thought of doing math without having to think. Now Leibnitz won. He's down to fairly recent times on that score.
But of course you all learn that way. You all learn math to do things without thinking. And it's the relatively rare student who while they're doing an algebra problem or a calculus problem actually pauses and starts thinking, what's going on here? It's not what you're taught to do, to pause and think.
You're taught to do it as fast as you can without thinking. Okay. That became part of the conflict between the Leibniz approach to calculus and Newton's. And remember, Newton never finished the great Geometria that was going to develop all of calculus starting from the most elementary geometry and doing all mathematics for exact mathematics.
That was a book, There are about six versions of the preface, and one or two versions of the introduction. Typical of a manic person to be doing that. A new version of the preface again, and again, but the rest of the volume is only described. In outline it's never done.
Any rate Take same book table of integrals, and it keeps going, the footnote is of course by Tom Whiteside but the 4th order, 3rd order curves, he's giving you what we call the integrals under them. One thing here and it's worth reading, I have enough time. The length, starting with the two points in here.
Transition to the method of fluctions. The length of the space described being continually, that is at all times given, to find the velocity of the motions at any time proposed. Number two the velocity of the motions being continually given to find the length of the space described in any time proposed.
And he gives an example that's not one I'm singling out. Just get down below. But since we do not consider the time here any further than acid is expounded And measured by an local motion. And besides where as things only of the same kind can be compared together.
And also their velocities of increase and decrease. Therefore in what follows I shall have no regard to time. Formally considered, but shall suppose one of the quantities proposed being of the same kind to be increased by an equitable fluxion, to which the rest may be referred, as it were to time.
And therefore by way of analogy it may not improperly receive the name of time. Whenever therefore the word time occurs and what follows, which for the sake of and distinctness I have sometimes used by that word I would not have it understood as if I meant time in its formal acceptation.
But only that other quantity by the equitable increase or fluxion where of time is expounded in measure. Now I am doing that in part to just show you how he treated time. It's just a parameter, its' the way you learn to use t as a parameter, as if it's time in modern calculus books, because of course you're picking up off of Newton and some of his solutions, which are very plain.
The other thing, though, was look at how carefully he's thinking this through. He's not 30 years old yet when he writes this. He's really thinking in detail about exactly what he is doing. And that permeates that book, okay? That's not a book of somebody working it out for the first time.
He's got total mastery of the juncture of 1671. Good. All right, any questions on the math? I mean, I made the main point, the Principia is not in this math, it's all in a Geometry with limits. He will occasionally fall back on infinite series, he will occasionally fall back on derivatives and integrals.
Because he can't do it Geometrically, and he will retread into something he can do. He is not reluctant to use whatever method he has to to solve a different problem. But he definitely comes to be uncomfortable with symbolic mathematics. The sort of phrase he uses is, very good, sometimes, to find the solution to a problem, then you have to figure out why it works.
That's my phrase and he doesn't say that. It talks about a method of analysis and synthesis. You really have to go back. The symbolic method can't stand on it's own.