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Bob3141592
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I have a problem with Newton’s reasoning in Lemma 28 (“There is no oval figure whose area, cut off by right lines at pleasure, can be universally found by means of equations of any number of finite terms and dimensions.”) It’s right after the start of Section 6 of the Principia, Book 1. I came across this lemma while reading Peter Pesic’s book “Abel’s Proof” about how equations of the fifth degree and higher cannot be solved in radicals. Newton’s Lemma is supposed to be a clear and elegant analog to this notion, going “far beyond the irrationality of pi” as Pesic describes it. (This description takes up the first part of Chapter 4, Spirals and Seahorses, in Pesic’s book.) Here’s a description of Newton’s argument, taken from the latter book:
“To use the term that Euler later introduced, the area of a circle is transcendental, meaning it cannot be expressed as the root of any equation of finite degree whose coefficients are rational numbers. At one stroke, Newton indicates that such magnitudes exist (because circles exist, and have areas), and also that there are infinitely many of them, since his proof is not restricted to circles but hold for any “oval” curve. His proof is a miracle of simplicity and power, for which he does not even bother to draw a picture or write down a line of algebra. It follows from a single brilliant contrivance. Inside the oval, pick any point whatever; let us call it the pole, P. Now let a straight line come out from that pole and rotate around it at uniform angular speed. Picture a clock hand that makes a complete revolution in one hour. Now imagine a point of light moving along that hand, starting from the pole and moving outward along the hand with speed given by the square of the of the distance from the pole to the point A where the hand intersects the oval.
Newton has set up a way of measuring the area of the circle, for each hour the hand sweeps through that area, and the moving point keeps track of that area because it is traveling with speed proportional to the area swept out. Here Newton is implicitly using his new calculus of motion, for he knows that, in an infinitesimally short time, the point travels a distance from the pole equal to the area the hand has swept out in that time. However, we don’t need to know anything about calculus in what follows. All that matters is that since, second by second, the moving point is registering the area that the hand sweeps out, we can measure the whole area of the oval merely by waiting until an hour has elapsed, and measuring the distance the moving point has traveled radially outward from the pole. For the two dimensional problem of measuring an area, Newton has substituted a one dimensional problem that gives the same answer: find the length traveled outward by the moving point during an hour.
The hand moves around uniformly, but the moving point speeds up and slows down in the course of each hour, in proportion to the square of the distance from the pole to the oval at any given moment. Each hour it returns to its initial speed of an hour before. If you were to watch the lighted point (or if you were to open the shutter of your camera and make a long exposure), you would see it move in a spiral, starting at the pole and making “an infinite number of gyrations” as Newton puts it. Now Newton applies a reductio ad absurdum: Suppose that it is possible to describe this spiral (and hence also the area of the oval) by some polynomial equation with a finite number of terms, f(x,y) = 0. For instance, Descartes showed that all the conic sections can be described by equations of the second degree, and those curves can be cut by a straight line no more than two times. Now Newton relies on the fact that an equation of finite degree can have only a finite number of roots, no matter how large. But the spiral in its “infinite number of gyrations” crosses the line an infinite number of times. Thus there should be an infinite number of intersection points, corresponding to an infinite number of roots of the equation. This contradicts our hypothesis that the equation has finite degree, and hence Newton’s conclusion follows: there is no such equation that gives the area of the oval.”
I have read the original text by Newton in the Principia, and it seems to me that the argument is much the same as Pesic describes it, though I find Newton’s archaic language a bit odd and hard to follow clearly. I understand about the area and the behavior of the spiral within the first hour. Subsequent to that, the lighted point sweeps out an irregular Archimedian spiral with separation equal to the area of the oval. But as the spiral “gyrates” on and on, it calculates not the area of the oval but, at first, twice the area of the oval, then three times the area, then four, etc. The area of the oval is determined within that first hour, and there the straight line intersects the spiral at most twice (trivially at the pole and at the full area point). That the line is allowed to continue sweeping out past the first hour introduces what seems to me a superfluous consideration. If the oval was a square the same steps would apply and it’d be hard to tell from the spiral what the shape of the generating figure was, yet the area of a square is expressible in radicals. It seems to me that Newton is saying that since x+1 and 2x+1 and 3x+1 and 4x+1 have an infinite number of solutions, that ax+b is not expressible in radicals.
Clearly that’s wrong, and I doubt that Newton and those that followed and Pesic missed this simple point. Since the spiral itself is not equivalent to a formula for the area of the oval but an infinite repeatition of the area, I think the deduced contradiction is invalid. So there must be something I’m overlooking or misunderstood. If anyone can point it out to me, I’d be very grateful. I’m interested in the Pesic book, but have suspended reading it until I clear up this point.
Again, let me thank the people in this forum in advance for your assistance and insights.
“To use the term that Euler later introduced, the area of a circle is transcendental, meaning it cannot be expressed as the root of any equation of finite degree whose coefficients are rational numbers. At one stroke, Newton indicates that such magnitudes exist (because circles exist, and have areas), and also that there are infinitely many of them, since his proof is not restricted to circles but hold for any “oval” curve. His proof is a miracle of simplicity and power, for which he does not even bother to draw a picture or write down a line of algebra. It follows from a single brilliant contrivance. Inside the oval, pick any point whatever; let us call it the pole, P. Now let a straight line come out from that pole and rotate around it at uniform angular speed. Picture a clock hand that makes a complete revolution in one hour. Now imagine a point of light moving along that hand, starting from the pole and moving outward along the hand with speed given by the square of the of the distance from the pole to the point A where the hand intersects the oval.
Newton has set up a way of measuring the area of the circle, for each hour the hand sweeps through that area, and the moving point keeps track of that area because it is traveling with speed proportional to the area swept out. Here Newton is implicitly using his new calculus of motion, for he knows that, in an infinitesimally short time, the point travels a distance from the pole equal to the area the hand has swept out in that time. However, we don’t need to know anything about calculus in what follows. All that matters is that since, second by second, the moving point is registering the area that the hand sweeps out, we can measure the whole area of the oval merely by waiting until an hour has elapsed, and measuring the distance the moving point has traveled radially outward from the pole. For the two dimensional problem of measuring an area, Newton has substituted a one dimensional problem that gives the same answer: find the length traveled outward by the moving point during an hour.
The hand moves around uniformly, but the moving point speeds up and slows down in the course of each hour, in proportion to the square of the distance from the pole to the oval at any given moment. Each hour it returns to its initial speed of an hour before. If you were to watch the lighted point (or if you were to open the shutter of your camera and make a long exposure), you would see it move in a spiral, starting at the pole and making “an infinite number of gyrations” as Newton puts it. Now Newton applies a reductio ad absurdum: Suppose that it is possible to describe this spiral (and hence also the area of the oval) by some polynomial equation with a finite number of terms, f(x,y) = 0. For instance, Descartes showed that all the conic sections can be described by equations of the second degree, and those curves can be cut by a straight line no more than two times. Now Newton relies on the fact that an equation of finite degree can have only a finite number of roots, no matter how large. But the spiral in its “infinite number of gyrations” crosses the line an infinite number of times. Thus there should be an infinite number of intersection points, corresponding to an infinite number of roots of the equation. This contradicts our hypothesis that the equation has finite degree, and hence Newton’s conclusion follows: there is no such equation that gives the area of the oval.”
I have read the original text by Newton in the Principia, and it seems to me that the argument is much the same as Pesic describes it, though I find Newton’s archaic language a bit odd and hard to follow clearly. I understand about the area and the behavior of the spiral within the first hour. Subsequent to that, the lighted point sweeps out an irregular Archimedian spiral with separation equal to the area of the oval. But as the spiral “gyrates” on and on, it calculates not the area of the oval but, at first, twice the area of the oval, then three times the area, then four, etc. The area of the oval is determined within that first hour, and there the straight line intersects the spiral at most twice (trivially at the pole and at the full area point). That the line is allowed to continue sweeping out past the first hour introduces what seems to me a superfluous consideration. If the oval was a square the same steps would apply and it’d be hard to tell from the spiral what the shape of the generating figure was, yet the area of a square is expressible in radicals. It seems to me that Newton is saying that since x+1 and 2x+1 and 3x+1 and 4x+1 have an infinite number of solutions, that ax+b is not expressible in radicals.
Clearly that’s wrong, and I doubt that Newton and those that followed and Pesic missed this simple point. Since the spiral itself is not equivalent to a formula for the area of the oval but an infinite repeatition of the area, I think the deduced contradiction is invalid. So there must be something I’m overlooking or misunderstood. If anyone can point it out to me, I’d be very grateful. I’m interested in the Pesic book, but have suspended reading it until I clear up this point.
Again, let me thank the people in this forum in advance for your assistance and insights.
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