Philosophy 167: Class 10 - Part 6 - Impact of Hard (Elastic) Spheres: Huygens' Solution, and Several Significant Consequences of It.
Smith, George E. (George Edwin), 1938-
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Impact. We of course know Descartes' impact. In the early 1650s, we don't have the original papers so we're guessing. But it looks like between 1652 and 1656, Huygens worked out the rules for impact. Not that the Cartesian rules for impact, the ones that go for what real odd and what happens to billiard balls, etc., that are very highly elastic, what they called hard.
He worked them out, didn't publish them, told people about them when he was in London. As a result, in 1668, two papers were submitted to the royal society. One by John Wallace and one by Christopher Wren. Now a moment about Christopher Wren, here. You have probably heard about Christopher Wren, but you probably heard heard of him as an architect.
Back in 1669, he had just been appointed as the architect to build the new St. Paul, which took the next 50 years of his life. It was finished in 1710, 40 years of his life, sorry. Christopher Wren, before that, was Professor of Astronomy at Oxford and a major figure in both mathematics, and in mechanics, and in astronomy.
So Christopher Wren and John Wallace both sent to the Royal Society papers announcing this is what happens on impact. Christopher Wren's were for perfectly hard bodies. John Wallace's were for perfectly soft bodies, bodies that don't rebound at all. They sent these two papers to Huygens to referee them.
Are these solutions any good? Cause they knew Huygens had the solution. He had told them that previously. So Oldenburg, Secretary of the Royal Society, sent these. Huygens sends back his own paper, expecting it to be published with the other two, announces Wren is right, Wallace is doing a different problem.
The Rail Society then publishes Wren and Wallace and don't mention Huygens, the referee in the paper he sent. Huygens took some offense. So, he then published a two page paper in French, Journal Francophone, announcing his results. At this point, Oldenburg, there's a correspondence I put under supplementary material, full of apology.
I didn't realize that you expected your paper to be published with, this is terrible. I really apologize, and Huygens writes back and says I'm so proud to be a member of your group, etc. So it's all gentlemanly, but clearly there's a lot of hurt feelings. So the Royal Society translates the the little two page French work into Latin, publishes it in Philosophical Transactions of the Royal Society with a page and a half explanation and apology to Huygens.
Now that is two pages long. It gives the modern solution not algebraically. That’s the textbook solution you learn in any class today. He gives it geometrically, but it’s exactly the same solution. He does not show how to derive it in this, he just states it. And then he gives four consequences of it, and they're stunning.
Number one, the quantity of motion which two hard bodies have may be increased or diminished by their collision contrary to what Descartes had said, conservation of motion, same all ways. No, no, false, can either increase it or decrease it. But when the quantity of motion in the opposite direction has been subtracted, there remains always the same quantity of motion in the same direction.
That's the first statement ever of our principle of conservation of momentum. Okay? Next, the sum of the products made by multiplying the bulk of each hard body into the square of its velocity is always the same before and after collision. That's the first statement ever of our principle of conservation of kinetic energy.
Third, a hard body at rest will receive more motion from another larger or smaller body if a third intermediately-sized body is interposed, than if would if struck directly. That's, we're talking about this. These are all the same size, unfortunately. But that's called Newton's Cradle, but of course Huygens and Rend deserve as much credit as anybody, and most of all if this third is their mean proportional.
What's that giving you? It's giving you a method of testing. What's it like as a method of testing? A very striking result, it's Galilean testing. Here's a consequence. Interpose a body that is the mean proportional, geometric mean between the two, you get more motion than if you don't interpose it.
Typical Galilean form of test, get a striking result out that you don't have to do quantitatively perfectly. And then the fourth, and notice that the interceding remark, and all of this, I am thinking of bodies of the same material, or else I mean that their bulk can be assumed from their weight.
By the way, the italics here are mine, this is obviously a slide I've used for different purposes and you'll see I've changed the translation. A wonderful law of nature which I can verify for spherical bodies in which seems to be general for all, whether the collision be direct or oblique, and whether the bodies be hard or soft, is that the common center of gravity of two, three, or more bodies always remains uniformly in the same direction, in the same straight line before and after their collision.
The center of gravity, whatever motion it has, keeps moving regardless of the interaction of the body. That's going to be the crucial principle in the Cornucopia in arriving at the Copernican system. So it's a very big deal in its own right. This is published in 1669. Newton definitely read it.
Okay, there's not a question about that.