Posts Tagged ‘Galileo’

Bounce, Part 6

January 11, 2010

Last time, we looked at what Galileo had to say about free fall. This time, we’ll take one more example from his dialog and try to squeeze a little moral out of it.

Galileo presents his ideas through the character Salviati, who explains them to his companions Sagredo and Simplicio. Salviati’s interlocutors raise all manner of objection to his theories, but Salviati answers them and convinces everyone of his point all the more surely in the process. One such objection is given by Sagredo, who doesn’t believe that the velocity of a falling object increases evenly with each second of falling:

So far as I see at present, the definition might have been put a little more clearly perhaps without changing the fundamental idea, namely, uniformly accelerated motion is such that its speed increases in proportion to the space traversed; so that, for example, the speed acquired by a body in falling four cubits would be double that acquired in falling two cubits and this latter speed would be double that acquired in the first cubit.

Sagredo is suggesting that rather than Galileo’s law

v \propto t,

that the velocity of a falling body increases the same amount each second, we should instead have

v \propto x,

the the velocity increases the same amount each meter the body falls. These are different hypotheses, and so we need to distinguish between them. Given that Salviati states he is not interested in examining the fundamental cause of gravity, and only in characterizing its behavior, there is only one way to do this – experiment.

Instead, Salviati offers the following retort:

…that motion should be completed instantaneously; and here is a very clear demonstration of it. If the velocities are in proportion to the spaces traversed, or to be traversed, then these spaces are traversed in equal intervals of time; if, therefore, the velocity with which the falling body traverses a space of eight feet were double that with which it covered the first four feet (just as the one distance is double the other) then the time-intervals required for these passages would be equal. But for one and the same body to fall eight feet and four feet in the same time is possible only in the case of instantaneous [discontinuous] motion;

What a strange counterargument! It makes absolutely no sense. Gaining an even increment of speed for each unit of time is a perfectly consistent mathematical law, and does not at all imply instantaneous motion. We can write this law as

\frac{dx}{dt} = c(x - x_0),

which implies that the distance fallen increases exponentially with time. This is completely contrary to observation, and it would be hard to build a unified mechanics like Newton’s that respects this law, but it isn’t logically impossible for the reasons Salviati cites.

And how do Salviati’s friends respond to this argument? Do they rip it apart, or restate their objection more clearly, or request further detail?

Sagredo replies,

You present these recondite matters with too much evidence and ease; this great facility makes them less appreciated than they would be had they been presented in a more abstruse manner. For, in my opinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion.

I guess it’s easier to convince the people you’re arguing with when they’re fictional characters you invented yourself!

Everyone makes mistakes, and they hadn’t gotten around to inventing peer review in the sixteenth century, so let’s forgive Galileo, and take a further look at this hypothesis.

Suppose we have a projectile with constant horizontal velocity and vertical velocity that changes according to the distance traveled in the vertical direction.

What happens if we shoot it up out of a cannon? We know from experience that the cannonball slows down, so it must be losing a constant amount of velocity for each unit height it gains.

The cannonball slows down its vertical velocity, but as it does so, its vertical position changes less. Since its vertical position changes less, the change in its vertical velocity slows down more. In fact, the cannonball asymptotically reaches a certain height above ground, and then stays there!

If the vertical height isn’t changing, then according to this law the vertical distance traveled is zero, and because vertical velocity only changes when vertical height changes, the vertical velocity stays zero. This cannonball would never come down.

On the other hand, if it were pushed down just a little bit, it would gain speed very rapidly, falling exponentially back toward Earth. The motion under Sagredo’s law is absurd, but I wonder why Galileo brought it up at all, only to miss the point.

Too much philosophizing can be dangerous, but this sort of philosophy – extracting results from speculative physical laws – is exactly what theoretical physicists do. The name of the game for a theoretical cosmologist, for example, is to come up with some crazy ideas about how the universe might work, the way Sagredo came up with an idea about falling bodies. Then, the cosmologist tries to work out the consequences of the theory, for example that cannonballs ought to hang in midair until a slight breeze comes along and gives them a downward tap, and they come plummeting back to Earth extremely quickly. If the theory doesn’t agree with observation, it’s wrong.

One difference between the Renaissance and Internet Age versions is that Sagredo’s and Salviati’s theories about falling are easy to test. The experiment doesn’t require any equipment. You just drop something. If you don’t have a thing, you can try jumping instead. But with advanced theoretical ideas, it can be very difficult to make the required observations. That’s why we need giant particle accelerators and kilometer-long interferometers and thirty-meter telescopes and ridiculously-good gyroscopes.

But another problem is that working out the consequences of modern theories is hard. We saw an example of Galileo failing to work out the consequences of a theory, but that was a simple mistake, and if someone had brought his attention to it, he’d have been able to fix it. Some of today’s new ideas about physics are so complicated that even if we can state the law (the equivalent of Sagredo’s idea about velocity being proportional to distance fallen), we may not know how to get to the conclusion (cannonballs hanging over our heads).

We’ve come long way since Galileo, figuring out lots of ways to check ourselves and test our ideas. But there’s a much longer way left to go.

Bounce, part 5

January 4, 2010

This post is a digression from the topic of the previous parts (1 2 3 4). We’ll move away from discussing how high a tennis ball should bounce when dropped on top a basketball, and into some metadiscussion of the arguments made in the first four parts. It’s a long post as well, but it’ll be good for you, because half the words are Galileo’s, not mine, and he’s a dude worth reading.

Last time, I cited Galileo as our source for understanding uniformly accelerated motion – the motion of a ball dropped or thrown in the air.

Before introducing his idea of what uniformly accelerated motion is, Galileo gives us an extended prelude. It’s long, but I think it’s worth seeing all at once, rather than piece-by-piece.

For anyone may invent an arbitrary type of motion and discuss its properties; thus, for instance, some have imagined helices and conchoids as described by certain motions which are not met with in nature, and have very commendably established the properties which these curves possess in virtue of their definitions; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exactly correspond with those properties which have been, one after another, demonstrated by us. Finally, in the investigation of naturally accelerated motion we were led, by hand as it were, in following the habit and custom of nature herself, in all her various other processes, to employ only those means which are most common, simple and easy.

For I think no one believes that swimming or flying can be accomplished in a manner simpler or easier than that instinctively employed by fishes and birds.

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody?

Galileo is mixing two approaches, and they appear to be intrinsically intertwined in his mind. The first is the ultra-skeptical pure empiricism viewpoint. This line of thought says that the only way to know about a thing is to confirm it by experiment. All scientific theories are to be tested against nature. If the theory and experiment agree, we fail to reject the theory. If the theory and experiment disagree, we reject the theory. Many modern scientists cite this as the true scientific viewpoint. (Note that from this point of view, you never confirm a scientific theory. Many scientists will agree with this – you never prove anything to be true in science. Also, I have called this viewpoint “empiricism”, a term which is sometimes used slightly differently in epistemology, where it refers to the belief that knowledge comes from sensory experience in general, rather than scientific experimentation in particular. Nonetheless, the cores of scientific and epistemological empiricism are similar.)

But, along with his statement that his knowledge of falling bodies comes from experiment, Galileo also has curious references to simplicity, in particular some out-of-place stuff about swimming fish and flying birds. This, to me, is the germ of a new idea – an idea that what we learn about nature ought to make sense to us on a deep level, once we’ve learned it. Greek philosophers (so I hear, not having read them) believed the Universe ought to make sense, and that they could therefore understand it with a priori reasoning. This is not quite what Galileo seems to believe. He holds himself responsible to experiment, unlike Aristotle, but I think that if experiment gave strange or unusual results that Galileo couldn’t understand, he’d be extremely dissatisfied. He feels a deep need to take the mathematical results, back them up with data, but then do even more. He needs them to make sense.

Two New Sciences is written as a dialogue (or, there being three interlocutors, a trialogue?), with Sagredo and Simplicio, two men who haven’t learned the new sciences, questioning Salviati, who has learned them and is explaining them to his friends. Galileo uses this device to explore intuition. He has Sagredo and Simplicio raise all manner of interesting objections to Salviati’s ideas, just so Salviati can find interesting answers to allay their unease. (This format is out of style in modern physics text, with rare exceptions like Spacetime Physics, a book I enjoy much more today than I did when first learning special relativity from it six years ago.)

For example, Sagredo thinks there is a problem with saying that a body dropped from rest has a speed proportional to the time fallen. He objects,

…we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.

He thinks there is a disconnect between the math and experiment, because the math says that when you drop something, it has almost no speed after falling a short distance, but Sagredo thinks that when you drop a heavy thing it starts falling quickly immediately. Maybe you don’t have this difficulty of intuition, but if you do, Salviati replies by appealing to an experiment.

You say the experiment appears to show that immediately after a heavy body starts from rest it acquires a very considerable speed: and I say that the same experiment makes clear the fact that the initial motions of a falling body, no matter how heavy, are very slow and gentle. Place a heavy body upon a yielding material, and leave it there without any pressure except that owing to its own weight; it is clear that if one lifts this body a cubit or two and allows it to fall upon the same material, it will, with this impulse, exert a new and greater pressure than that caused by its mere weight; and this effect is brought about by the [weight of the] falling body together with the velocity acquired during the fall, an effect which will be greater and greater according to the height of the fall, that is according as the velocity of the falling body becomes greater. From the quality and intensity of the blow we are thus enabled to accurately estimate the speed of a falling body. But tell me, gentlemen, is it not true that if a block be allowed to fall upon a stake from a height of four cubits and drives it into the earth, say, four finger-breadths, that coming from a height of two cubits it will drive the stake a much less distance, and from the height of one cubit a still less distance; and finally if the block be lifted only one finger-breadth how much more will it accomplish than if merely laid on top of the stake without percussion? Certainly very little. If it be lifted only the thickness of a leaf, the effect will be altogether imperceptible. And since the effect of the blow depends upon the velocity of this striking body, can any one doubt the motion is very slow and the speed more than small whenever the effect [of the blow] is imperceptible? See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary. (brackets added by translator)

I get the feeling, while reading this passage, that Galileo cites this experiment simply because it gives him pleasure to do so. But in this case, even the experiment is not enough for him. He continues

But without depending upon the above experiment, which is doubtless very conclusive, it seems to me that it ought not to be difficult to establish such a fact by reasoning alone. Imagine a heavy stone held in the air at rest; the support is removed and the stone set free; then since it is heavier than the air it begins to fall, and not with uniform motion but slowly at the beginning and with a continuously accelerated motion. Now since velocity can be increased and diminished without limit, what reason is there to believe that such a moving body starting with infinite slowness, that is, from rest, immediately acquires a speed of ten degrees rather than one of four, or of two, or of one, or of a half, or of a hundredth; or, indeed, of any of the infinite number of small values [of speed]?

Here we see the second approach to nature. The idea that, once we’ve formulated a theory and tested it, we’re still not done. We need to reason about it, too. We need to go back, take the solution, and make it ours. We need to convince our grandmothers, who don’t know math, that this is the way it ought to be. And both these processes are intertwined. You can use the idea that nature ought to be simple to figure out what the laws are, but if you do, you’re still subject to testing them by experiment. Conversely, you can use experiment to figure out the laws, but if you do, you’re still subject to figuring out why things came out that way.

Galileo is the earliest source I’ve seen with this new, sophisticated attitude. Naturalists wanted to observe, discover, and document what happened around us. Philosophers wanted to talk about it in the abstract and explain its deeper logic. But Galileo wanted to do both. And it’s only when you do both that you’ve accomplished the real goal – understanding.

I’m not saying this attitude sprung up in Galileo’s work with no precedent, but I do think it’s clearly evident here, and since Two New Sciences is a landmark work in terms of the physical ideas it presents, it’s important to examine in terms of the philosophical ones is presents, too.

This Galilean principle still guides us today. Science isn’t about testing hypotheses and controlling experiments and statistical significance. Science is about figuring things out. The methods of modern science evolved over time as the problems scientists dealt with demanded them. (A great deal of statistics was invented specifically to study genetic inheritance, for example). Galileo didn’t have our textbook scientific method, but ultimately he didn’t need it to make great progress.

Today we need things like careful laboratory conditions and error propagation formulas to keep us from screwing up when things get tricky and hard to interpret. But the core of my world outlook, which I am not afraid to claim is also the core of the scientific one, is that you are just trying to figure things out, subject to checking what really happens, and then, once you do that, trying to understand.

Next time, I’ll take a look at one of Galileo’s arguments that didn’t work. That’s the other thing about science that I like. Nobody’s perfect, and you’re expected to screw up at least once in a while.

Bounce, part 4

January 2, 2010

Previous parts: 1 2 3

Last time we made progress on figuring out how high a tennis ball can bounce in the classic experiment where we drop the tennis ball on top a basketball. We didn’t find the answer, but we said that if the tennis ball picks up a speed v in falling, then immediately after bouncing off the basketball, it could have a maximum upward speed of 3v.

Today we want to figure out what that means in terms of how high the tennis ball will bounce. It turns out that the tennis ball does not bounce three times as high as it started when it rebounds with three times the speed. In fact it bounces much higher.

After bouncing off the basketball, the tennis ball rises, but slows down under the influence of gravity until it comes to a stop at the top of its trajectory. To understand how high it goes, we must answer the question, “what does the influence of gravity do to the motion of the ball?”

One of the first people to understand this question and its answer was Galileo (although several people came to the correct conclusion before him). We’ll look at a few passages of his famous book, Dialogue Concerning Two New Sciences. (specifically this part)

Galileo begins by stating that he thinks “uniformly accelerated motion”, the motion of a tennis ball thrown into the air, should be very simple.

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner.

In other words, the way the speed of a falling body changes shouldn’t depend on how high it is, or how long it’s been falling, or how far it’s fallen. It should depend on nothing at all – be always the same.

This may be a lot to swallow, but let’s look at one good reason (not due to Galileo) that we might expect the way gravity acts on a falling object not to change with how high the object is above the Earth’s surface. The radius of the Earth is very large compared to the heights we throw things. We expect that if the effects of Earth’s gravity do change with your distance from the center of the Earth, they ought to do so on a distance scale roughly equal to the radius of the Earth.

That is, if you want a significant difference in the force of gravity, you ought to change your position by something significant compared to the radius of the Earth, since it defines the only natural length scale in this problem. The radius of the Earth is roughly six million meters, so throwing a tennis ball up in the air six meters is completely negligible. We could calculate the effects of gravity using Newton’s gravitational law, but that is unnecessary. Any other reasonable gravity law ought to work out basically the same. Near the surface of the Earth, your height should not affect how gravity acts on you.

This is only one part of what Galileo said. For example, he also believes that how fast an object moves should not affect how gravity acts on it. This belief may have been stimulated by the relativity principle – that all laws of physics should be the same, even when you’re moving. Relativity does not absolutely preclude a force that depends on velocity, though (magnetic forces do this), but velocity-dependent forces are not as simple as velocity-independent forces, and for the time being Galileo is guessing that the way gravity acts ought to be very simple.

We continue with the G-spot’s wise words:

A motion is said to be uniformly accelerated, when starting from rest, it acquires, during equal time-intervals, equal increments of speed.

This is Galileo’s working idea of how things fall. If you drop something, and at the end of one second it goes speed v, then at the end of two seconds it will be going 2*v, and at the end of three seconds 3*v, etc. A plot of speed over time, if we drop an object from rest, should look like this:

The speed of a falling tennis ball, starting from rest.

This plot shows the speed of a falling tennis ball. The tennis ball is dropped from rest, and so starts at speed zero. Gaining equal speed in each moment of time, the speed is directly proportional to time.

Now that we have a theory for what the speed of the tennis ball does, we should be able to figure out how high it goes. The tennis ball reaches its highest height when its speed is zero, so we simply need to keep track of its speed until that speed falls to zero. If we know how fast it was going and for how long, we should also know how far it traveled.

I’ll paraphrase Galileo’s arguments here rather than quoting them, since he does not directly answer our exact question. The relevant pages are 171 – 178.

First, let us suppose it takes the tennis ball a time t to fall before bouncing, and it acquires speed v in that time. We know it bounces back up with speed 3v. It loses speed in the same way it gained speed – the same amount per second. So after a time t, the ball loses speed v, and is down to moving at speed 2v. The ball comes to a stop at the height of its trajectory after a time 3t.

To summarize, if the ball gains and loses the same amount of speed in any moment of time, then if it two balls bounce upward, one three times as fast as the other, the fast one will take three times as long to get reach its apex.

The distance the ball travels just speed * time, which is the green area shaded in the previous drawing.

Here is a plot of the speed of the ball as it rises:

tennis ball's bounce

The tennis ball's return trip. This time it begins going quickly, three times as fast as before, and slows down. It takes three times as long to reach its peak as it took to fall.

It rises three times as long as it fell, and the distance it rises is purple the area in the above chart. Laying the two plots together, we see that the purple area is nine times as large as the green one – three times taller and three times wider.

rise and fall comparison

The green area represents the distance the tennis ball fell (see first figure). The purple area is the area the tennis ball rises after bouncing off the basketball. The tennis ball rises nine times as high as it was dropped from.

Now we have our first answer to how high a tennis ball can bounce when dropped on top a basketball. It can bounce nine times as high, when we make the following assumptions:

  • When things bounce off the ground, they change their direction keeping exactly the same speed and hence bounce back to the same height. (first post).
  • A basketball is so much bigger than a tennis ball that it essentially acts as the ground – the tennis ball bounces off just the same as it would bounce off the ground. (third post)
  • To understand the way something in motion works, we can imagine we are moving alongside it at the same speed so that it isn’t moving from our point of view, and understand it that way. Then we can imagine going back to the frame in which the thing is moving and translating over our new knowledge over. (third post)
  • Gravity pulls an object down such that it gives it the same additional amount of speed in each moment of time. (this post)

My original claim was that I could have understood all these ideas as a child. I think that’s right. I was a pretty bright kid, and if someone had sat down to explain this reasoning to me, and answered my questions, I think I’d have gotten it. But I also hope I’d have realized there’s a problem. When you actually do the experiment, the tennis ball doesn’t bounce nine times as high, or anywhere near that. Three times as high is pretty good for this experiment. So I’d like to think I’d have noticed that, and asked for an explanation of the discrepancy.

We began to discuss this in part two, where we looked at why things bounce to a lower height than they’re dropped from. The assumption about reversing direction and speed when bouncing is simply not correct. It is also not correct to assume that the basketball is so much larger than the tennis ball that it acts like the ground, but this is a smaller source of error. It isn’t true that gravity is completely uniform, either, or that the only influence on the falling ball is from gravity. We’ll look at these things in more detail in a later post.

Before doing that, though, the next post or two will continue looking at the passage from Galileo. This passage isn’t interesting to me simply because it is an early source of someone understanding this fairly simple problem. It’s interesting because it’s an illustration of Galileo laying down a more sophisticated understanding of how we can understand nature. I want to look at what Galileo did and didn’t know, but also at how much he understood about what he did and didn’t know, and how he came to his conclusions.

There’s also a very surprising and egregious logical error in the passage, so we’ll talk about that, too, before returning to the tennis ball a little down the line.

The Renaissance Man Uniform Gravitational Acceleration SMACKDOWN

October 9, 2009

Matt at Built On Facts posts about coriolis forces, and points out that a falling body is deflected by them one eighth as much as one tossed from the ground to the same height, and that they’re deflected in opposite directions. Here’s my attempt to explain this intuitively.

This makes me think of the competing da Vinci – Galileo laws for bodies (not their own I hope) falling freely under gravity. They stated their rules in the same basic way. I remembered these laws from watching The Mechanical Universe in high school – before taking physics from the real life version of David Goodstein three years later.

da Vinci said (or so I hear, I never met the guy) that if you fall one unit of distance in the first unit of time, you’ll fall two in the next unit, three in the one after that, then four, etc. So if you fall 5 meters in one second, in the next you’ll fall another 10 for 15 meters total.

Galileo said almost the same thing, but with odd numbers. If you fall one unit of distance in the first second, then in the second you fall three, then five more, then seven, etc. So if you again fall 5 meters in one second, in the next you’ll fall another 15, for 20 total.

Galileo was right; da Vinci wrong. But let’s not screw over our primitive-flying-device-making friend with such a cursory examination. They’re both awesome dudes, as Leonardo’s testudine counterpart would say.

Galileo was right because acceleration is constant, so the distance fallen is proportional to the square of the time. Adding Galileo’s odd numbers gives a square number. 1+3+5 = 9, for example. This is easy to see from a picture.

Each new section adds the next odd number worth of dots, and takes you to the next bigger square number when counted as a whole.

Each new section adds the next odd number worth of dots, and takes you to the next bigger square number when counted as a whole.

da Vinci, instead of the square numbers for total distance fallen, gave the triangular numbers. 1 + 2 + 3 = 6, which is triangular. This has its own picture.

According to da Vinci, each new row is how much you fall in one additional second.

According to da Vinci, each new row is how much you fall in one additional second.

da Vinci’s fub may have been in misunderstanding the relationship between speed and distance. If da Vinci’s rule had been giving the speed at the end on each second, rather than the incremental distance fallen, he’d have been right. If you’re going 10m/s after one second, you go 20m/s after two, and 30 m/s after three, etc. The problem is that you can’t find the distance traveled in a second by taking the speed at the end of that second and multiplying by time. If you do that, you get only an approximation to the correct integral, like this:

Dont worry about the numerical details.  I stole this from the internet somewhere.  da Vincis law overestimates distance fallen every second by assuming your speed at the end of the second was you speed for the entire second.

Don't worry about the numerical details. I stole this from the internet somewhere. da Vinci's law overestimates distance fallen every second by assuming your speed at the end of the second was you speed for the entire second.

It’s possible that da Vinci was actually right on about the kinematics, but that he made a mathematical error in reporting his result. I wanted to follow up on this, so I checked online to see precisely what Leonardo said. I did not succeed. Fritjof Capra’s book quotes da Vinci:

The natural motion of heavy things at each degree of its descent acquires a degree of velocity. And for this reason, such motion, as it acquires power, is represented by the figure of a pyramid.

But when I search online texts of Da Vinci’s notebooks, I can’t find this passage. I can’t find the relevant passages in my Dover copy of Richter’s translation, either. In fact, I can’t find this passage anywhere else on the entire internet, except one book that doesn’t cite the source. So I’m not sure what to make of this. da Vinci’s writings on falling bodies must be somewhere, if we know about them. But as of now I’m still uncertain. Based on the preface to my translation of the notebooks, it looks like they decided to omit some of Leonardo’s physics, since that is obviously unimportant and uninteresting to readers of his notebooks.

Let’s assume Leo had the right idea, but brain farted on the integration thing. Considering how clever Da Vinci was, his mistake is very surprising, because his law is not only empirically wrong, it is logically impossible.

To see what I mean, let’s carry out Da Vinci’s argument a little further. According to his rule, in four units of time you fall 1+2+3+4 = 10 units of distance. But the choice of how long a unit of time is was arbitrary. So let’s do it again, but consider the unit of time to be twice as long as it was previously. We’ll call these “shmunits” of time. In one shmunit of time, you have to fall three units of distance to be consistent with the first calculation. Then you fall six units of distance in the second shmunit of time, because the second has you falling twice as far as the first. After two shmunits of time, you fall a total of nine units of distance. But we already said that with the same law you fall ten units of distance! Surely if Leonardo had considered his law carefully he’d have seen this error, right?

Unless it’s not an error. What if Leonardo actually meant that you have to take the limit as your unit of time becomes infinitely short? In that case, Leonardo’s law

distance \propto t(t+1)

can simply be reduced to the correct law

distance \propto t^2.

Could this really have been what Leonardo had in mind? I think it’s possible, but not likely. The Greeks explored the basic ideas here. They approximated \pi using the method of exhaustion, and Archimedes is said to have been doing what amounted to integral calculus. If Leonardo was aware of this research, he might have stated such a law accurately. But it seems far-fetched.