Prior to 1679, November 24th 1679, when Hooke sends his letter, first of the letters, I think it's safe to describe Newton's work in both mechanics and in astronomy is largely dabbling. Dabbling in response to reading the dialogue on two world's Chief to Cheek World Systems. And with his rabid anti-Catholicism wanting to establish the Copernican system if only to justify Galileo in the face of the church.
And then Descartes' Principia which he, in spite of the influence it had on him, he really did not like the book very much at all. So those two produced some of the stuff we looked at last week. Then perhaps we're not sure because he doesn't date things, the 1669 papers that appeared in field transactions on impact by Wallace, Ren, and Huygens, and finally Huygens' horologium oscillatorium.
All of those he's responding to dabbling. Usually, one could argue simply looking at them as interesting mathematical problems but not trying to do much in the way of sustained Physics. Beyond in some respects what he had done when he was young, in the waste book. He always was fascinated by any mathematical problem I was at a dinner last night where Frank Wilczek started saying, how could Newton have spent so much time calculating when various prophecies happened by working backwards from the stars to date things?
These are his Chronicles, and you know, I said to him right away, you don't understand. Newton did the same thing with Solomon's temples. Once an argument broke out about it at a lunch at Trinity College, he figured out what from the bible what the dimensions etc. of Solomons temple have to be, and that's a document we had a Dinner Institute is carefully working things out.
He loved problems. And very often, he gets into something merely because of a problem. At any rate, from here on, we're worried about 1679 and after that. Everything in this course, so far, has just prepared us to get to 1679. Three years, 1676 to 1679, fill transactions had ceased publication after Oldenburg died.
And I guess they came to realize how much effort Oldenburg put into correspondence, which is like 16 volumes of his correspondence. Some immense number, and ordered to solicit papers for fill transactions, particularly high quality papers. So Hooke finally was given the responsibility of resurrecting fill transactions, so he sent out a whole series of letters, including this one of 24 November, 1679, which reached Newton not too long after he returned.
To Cambridge from Wilsthorpe, after settling his mother's affairs when she died in 1679. Hooke poses seven different problems. I'm not going to go over stuff you can just read, but there are seven distinct problems that he poses that he hoped Newton would do something about. One is direct motion by the tangent and attractive motion toward the central body.
His own so-called, law or cause of springiness, we call it Hooke's law. A French hypothesis, the work that Cassini and Romer were doing on trying to locate longitudes and doing mapping. A question of getting the latitude difference between London and Cambridge. La Hire's beautiful book on conic sections, which you'll see referred to in the Principia.
It's a book that Newton admired a great deal. And finally this remark Flamsteed by some late perpendicular observations has confirmed the parallax of the orb of the Earth which, of course, he had not. I'm not even sure Flamsteed said that. I suspect the most he said was maybe but be that as it may.
So Hooke sends this letter, 24, November, 1679. Asking Newton to submit something to the fill of transactions and giving him a whole range of topics that he thought Newton might respond to. And what Newton does, I'll flash down one step and you see the opening sentence here. I'm glad to hear that so considerable discovery as you made of the Earth's annual parallax is seconded by Mr. Flamsteed's observations.
Remember in 1674 Hooke had published a book announcing that he had seen the observations. So Newton is showing him some credit. Hooke doesn't take credit in the letter, but Newton shows him. In requital of this advertisement I shall communicate to you a fancy of my own about discovering the Earth's diurnal motion.
So look, here again what Newton is preoccupied with is establishing some aspect of Copernicanism, this time the rotation of the Earth. And it's important because you'll see later, much later, next spring Huygens in 1684 is equally preoccupied with trying to establish the rotation of the Earth. It simply wasn't viewed as having been established at the point.
So Newton's proposal, it's become, it's really far more important than you tend to learn about in school. It's now called Coriolis forces. The idea's fairly simple. I put the tower at the equator, I would have been better off when I made this silly drawing, putting it at 45 degrees or right a little more than that to represent London.
But the idea is, if the Earth is rotating around the axis, which is horizontal here, and you drop an object from a tower, at the moment you drop it, it's moving laterally faster than the base of the tower. So instead of falling behind the tower, the way proponent's said, or falling at the base of the tower, which is what Gassendi said he had actually measured dropping on a mass.
Newton's pointing out, if the Earth's rotating it's gotta land ahead because it's going faster speed than the base laterally. And by the principle of inertia, of course, straight line motion instead of circular motion, that means the greater velocity's gonna bring it down. If you guys don't picture this just sometime.
You probably can think this through even without having done it. But get on a merry-go-round, and try to toss a tennis ball back and forth. It's very very hard to do. If you're on the outside, you gotta start throwing it behind the person. And if you're on the inside, you've gotta start throwing it way ahead of them.
And that's the principle, Newton proposes it. And says, and the striking thing, I'll get to that in a moment. I just want you to realize it's a sensible principle. Hooke comes back right away and says it should be both to the south and the east if we drop it in London, not just the east.
And he was right about that because of the angle that the tower takes relative to the axis. But the real point is it actually is in principle, if the Earth is rotating, it in principle, the body ought to land ahead of the base. And again, I'll take just a moment, I think I've said this earlier in the course, but I like to tell people.
The reason I say it's enormously underestimated how important this is to call it Coriolis forces, that's a name from the 19th century, They totally dominate our weather and all motion in our seas. All things like gulf stream, et cetera. The fact that you've got faster motion on the surface than you do at depth.
Faster motion at high altitude than you have near the surface of the earth is why we have storms, why we have all these lows and highs. All of that is coming off of Coriolis forces. They're not forces, of course. There appear to be forces and that's not a trivial manner.
Unfortunately, couple of things about this, well I'll come to whether you can do the experiment in a moment. This does not appear in the Principia because Newton comes to realize that to do, to actually get a reliable effect you're going to need a much, much higher tower and you're going to need to control for air resistance.
At any rate, the striking thing, the reason I included this letter, is when he starts describing the actual experiment and how to do it, it's kind of remarkable. So typical of him. He doesn't just say, drop it off of a tower and see where it lands. Let's start at the bottom of the left-hand column.
The advance of the body from the perpendicular eastward well into the descent of 20 or 30 yards will be very small, and yet, I am apt to think it might be enough to determine the matter of fact. Suppose then, in a very calm day, a pistol bullet were let down by a silk line from the top of a high building or well.
The line going through a small hole made in a plate of brass or tin, fastened to the top of the building or well. And that the bullet, when let down, almost to the bottom were settled in water so as to cease from swinging and then let down further on an edge of steel, lying north and south, to try if the bullet, in settling thereon, will almost stand in equilibrium.
If yet with some small propensity, the smaller the better, decline to the west side of the steel as often as it's so let down thereon. The steel being so placed underneath, suppose the bullet be then drawn up to the top and let fall by cutting, clipping, or burning the line of silk, and if it fall constantly on the east side of the steel, it will argue the diurnal motion of the earth.
You see he's really designing the bloody experiment in a great deal of detail. This is absolutely typical of him when he gets to any experiment. He doesn't just say, go do the test. He starts designing it in full detail. At any rate, the other thing to notice here.
Hook's letter is November 24th. Newton's reply is November 28th. The mails were not a whole lot worse than they are now. At least, between London and Cambridge. I always laugh at that. The mails tended to be overnight from London to Cambridge. And a day and a half later it gets there.
Of course, there wasn't much mail, not much junk mail at all. Hooke replies to this drawing, well he replies in two respects. He replies to the proposal saying he announced it before the Royal Society and they got very interested, they were quite excited, all this stuff. Then he said by the way, the drawing you do here, that's wrong.
It's not gonna spiral down unless there's resistance. It's gonna do something more like an elliptoid I think is the word he uses. Yes, the motion has to be a kind of elliptoid. And at which point he sends this back to Newton. And Newton replies, and this is a case where I'm having to show you the original letter to make a point.
What Newton then does, if you've read the letter, it's available to you on Trunk, so I'm not gonna worry about the details. He says, you're right, it's not gonna be a spiral like that, which he'd done. He's thought through. What's the motion going to be? Suppose at point A, you have a horizontal motion, this is Galileo's projectile problem, and you have constant gravity all the way to the center.
What's the path gonna be. Well he's drawn a line down to a point o, that's the nearest you get to the center. Comparable to parent helium, and then notice what he's done. He seems to have traced that line, flipped it and continued the line to match the first half.
So it's the mirror image of the first half and you can see how heavily this is drawn through here. And the consequences he's almost got the right solution but the point of contact, H, is too far, it should be about 200. I'll give you the number because he calculates it in the Pricipia, 203.8 degrees.
And this is around 220 degrees if you use a protractor. What's he done, it's fascinating. He's worked out the right account, at least qualitatively and pretty well with curvature. Down to point c. But then when he traced it he just was a little bit sloppy and threw the line off.
He didn't actually do the full analysis to see where the angle would reach a maximum. But you see what it does, it keeps doing lobes of this sort, and his description of it is it alternately is drawn toward and goes away. Again, I wanna use his own words, I think I have his own words.
Yea, the vista centrifica and gravity are alternately over balancing. So gravity pulls it in but it's going so fast by the end that it goes back out, the vista centrifica pushes it out. Then gravity out here becomes dominant, pulls it back in, and it keeps going. This is the solution to the problem that Mersenne posed that everybody was trying to get.
What trajectory does a projectile, in the absence of air resistance describe, if gravity is uniform all the way to a center? This is actually as near the solution as anybody had ever come. Nobody had even come close to it before this. Newton does it analytically in the Principia and gets the modern solution to four significant figures.
So it's fairly striking, to see it. And you now see why I had to show you the original. You couldn't see in the printed version that that drawing was not what it appears to be in the printed version. It's a case where he has tried to complete the drawing without, it's not that easy to line up a curvature like that.
How he got the curvatures, that's a perfectly interesting question, but we have a pretty good idea. Since he knew a lot about curvature. And I'll show you a little bit later, he knew enough about curvature that probably he did this. The figure here I pulled out of an article by Bruce Breckenridge, but the person who came to realize this is Michael Nouwenberg, a physicist who late in his career turned into a Newton scholar at Santa Cruz.
In the University of California at Santa Cruz. But Michael should get the credit for this. If he were in the room and I didn't give him credit, I would never hear the end of it. At any rate, that's the last letter Newton sends in this, and I think it's the last letter he ever writes Hooke in Hooke's Life, Hooke died in 1703.
Now one of the things that probably perturbed him was Hooke telling him in the letter that he had told the Royal Society about Newton's mistake on the original description. And, at the very least, Newton comes back and gives him a crack one. But now, I'm gonna do these in sequence, but the 6th January reply The preceding letter, the one I just showed you.
The form on, is December 13th. So Hook replies December ninth to Newton's November 23rd letter, and then. Newton comes back December 13th. So they're pretty regular in there until a month delay, and then the sixth of January. So let's just see, it's the principle challenge I'm gonna be concerned with here.
In the 24th of December, the way he posed this challenge was particularly, if you would let me know your thoughts of that hypothesis of mine, that's my answer. Of compounding the celestial motions of the planets, of a direct motion by the tangent, and an attractive motion towards the central body.
Notice something right away, two motions. We don't think of circular motion as composed of two motions. Newton thought of it as a tendency, a connodice to recede. Hook's posing it as there are two separate motions here compounding to generate the circle. That's a striking proposal. December 9th, but as to the curved line which you seem to suppose it to descend by though that was not then at all discovered of vixed a kind of spiral which after some few revolutions, leave it in the center of the earth.
My theory of circular motion makes me suppose it would be very differing and nothing at all akin to a spiral but rather a kind of ellipsoid. By the way if you're concerned about spelling and grammar you should realize English was not an established spelling and even grammar compared to the Latin.
All these people's Latin is really beautiful and their English is all over the place including spelling all over the place. But there were dictionaries in Latin. There weren't dictionaries of English at the time. Those came later. All right, then the key thing in the sixth of January, and even this I'm not quoting in full.
Your calculation of the curve by a body attracted by equal power at all distances from the center of such, as that of a ball rolling in an inverted concave cone, is right. Isn't that nice, he tells Newton that calculation is right. And the two augites, those are the two extreme points, will not unite by about a third of a revolution.
But my supposition is that the attraction is always in the duplicate proportion to the distance from the center reciprocal. And consequently that the velocity will be in a sub-duplicate proportion to the attraction. And consequently, as Kepler supposes, reciprocal to the distance. Remember, Kepler didn't exactly suppose that. He supposed that at one point, velocity is proportional to one over the distance but then concluded that only holds at the extreme points for the area rule.
Hook is saying to the contrary. And that with such an attraction, the augs will unite in the same part of the circle. And that the nearest point of access to the center will be opposite to the furthest point. Notice he's not saying it's an ellipse, but he's saying it's symmetric with a major axis like that of an ellipse.
I'm skipping some stuff that I probably shouldn't have, but I only had so much space on the page. What I mentioned in my last concerning the descent with the body of the Earth was but upon the supposal of such an attraction. Not that I believe there really is an attraction to the very center of the Earth, but on the contrary I rather conceive that the more the body approaches the center, the less will it be urged by the attraction.
But in the celestial motions of the Sun, Earth, or central body are the cause of the attraction, he said. In the celestial motions, the Sun, Earth, or central body are the cause of the attraction, and though they cannot be supported mathematical, they cannot be supposed mathematical points yet they may be conceived as physical and the attraction at a considerable distant maybe computed according to the former proportion as from the very center.
This curve truly calculated, will show the error of those many lame shifts made use of by astronomers to approach the true motions of planets with their tables. Now all the tables at the time were based on ellipses, so I take them here to be questioning whether they're actually ellipses.
Okay? Newton does not respond to this. Newton never responds again, as they say. So we get this. I did skip one. The part I skipped in the middle there is actually worth mentioning. Assuming I can find it right away, yes. Rather a kind of ellipsoid. At least the falling, yes.
That the gravitation of the former center remained as before and that the globe of the earth were supposed to move with the diurnal, I'm sorry, I'm not finding the spot I want. Sorry, I looked at the wrong letter. The finding out of the properties of a curve made by two such principles, the two motions, will be of great concern to mankind.
I dropped that in the middle of that quote. So he's saying this is a very, very big deal, okay. Newton doesn't respond. And finally, still wondering why Newton's not responding, eleven days later here, Hook writes him again. It now remains to know the properties of a curve line, not circular, not concentric, not concentrically.
Made by a central attractive power which makes the velocities descend from the tangent line are equal straight motion at all distances in a duplicate proportion to the distances reciprocally taken. I doubt not but that by your excellent method you will easily find out what that curve must be.
And its properties. And suggest a physical reason of this proportion. Okay? Of course, the excellent method, everyone knew that Newton had developed a new algorithmic method for solving problems. But they didn't really know what it was because one set of papers was circulating and then maybe 10 or 15 hands.