Versand, in particular, seems not to have been that much persuaded by any inclined plane experiment. So what he started doing was trying to do direct vertical fall experiments. The first thing he does is he takes a wall, takes a ninety degree pendulum holding out here, same size sphere here and he lets them go at the same time.
Seeing how they hit together and he thinks doing that and knowing that period of the pendulum, he can get the distance of fall in the first second. He has the distance of the fall in the first second, he can do experiments, because it should be three times as long in the second second, five times as long in the third, etc.
Okay, this leads him to conduct some vertical fall experiments in 1634, this is before two new sciences appears. Where he puts in print the following, three feet in the first half second, twelve feet in the first second. Total, that's right. Square going up half second to one second.
48 feet in two seconds, 108 in three seconds, 147 in 3.5. Those are exact agreements his actual data are slightly off, instead of 108 it's 110, feet at 147 in 3.5 seconds. The highest height he could drop from was about 150 feet and that's only 3.5 seconds but he concluded at that point that maybe the 1357 actually holds.
Then he, this gets published, and he decides he wants to redo all of this, so he gets permission from the Vatican to go to the top of St. Peter's Stone and drop objects from 300 feet.
Okay, and does the experiments over again, and concludes well, it's at least roughly right, but when you look carefully, it should be landing in five seconds and is varying between five and six seconds and the conclusion is, therefore, all you can really say with confidence about Galileo's uniformly accelerated motion is it's at least an approximation.
To what would happen in the absence of air resistance, okay, and that's in the mid-1640s. Separate from that up in Bologna, wonderful town, is a guy named Riccioli, Giovanni Battista Riccioli and he decides well, I should back up. He takes on in 1640 or so he's a professor, he's a Jesuit professor at the University of Apollonia.
For those who don't know, because for some reason it's not as famous as Oxford, but it and Oxford vie for one another to be the oldest university in Europe and it's a marvelous university. I remind you it's where Copernicus learned his advance astronomy etc. It's a great university and at that time it was a great university.
Richioli is a professor there and he's famous now because he came out with the last defense of Ptolemy called the New Alma Guest and I'll be Quoting from it, and therefore he's laughed at because he's still defending Ptolemy in 1651. That's bad press, what he decided to do was to do his honest best to check everything Galileo said and anywhere the empirical evidence was adequate he would go with Galileo.
Anywhere it's not, the traditional position held sway, so he tried very, very hard to be fair to Galileo even though he's represented as being an advocate of Ptolemy. So what's he do? The first thing he decides to do is he's got to measure time really accurately. Mersenne had proposed the length of a one second pendulum but it's not at all clear how accurate it really was and Mersenne himself had pointed out there was something really odd because when you do this experiment dropping, the pendulum lands first.
Not the object falling directly and thought that very puzzling. We'll come back to that. Cuz that's what starts Huygens off on this whole thing is trying to figure out what was going wrong with Mercenne. So I'll I'll now read, what Riccioli did was to take a pendulum that he guessed.
Three old Roman feet, four inches in length, with a one pound bob and he calculated, excuse me, he measured the number of arcs in 21,660 sidereal seconds. And it didn't agree with what it should to be a sidereal second. Actually it's a mean solar second but let me not worry about that now.
Second example, a pendulum with three old Roman feet, four inches with a different size eight ounce iron sphere. This time he did it for a whole day 87,758 arcs in 86,400 seconds. Now I hope you're asking, did they really count 87,000? So here's what he says. This is translation by myself and Judy Nelson.
I set up nine companions. You wouldn't like to be one of Riccioli's friends right? Well instructed in this matter who almost all publicly practice philosophy or theology or mathematics, so that they succeeded each other in the counting after about every half hour. And in the year, 1642, from noon on April 2nd to noon on April 3rd, we maintained a count of simple vibrations, whose number from the pebbles thrown in the vase every 60 vibrations, was found to be 1466 sixties and in addition 38 vibrations.
But a day of the primum mobile, that's the sky, contains 1440 of it's own minutes. The solar day indeed is 1444 prime mobile minutes. Therefore such a pendulum, leave that alone for the moment that's a sidereal, therefore such a pendulum in one day of the Primum Mobile completes 16 times 1462 vibration in addition 38 vibrations when it ought to complete only 1440.
If a single simple vibration corresponded to one second therefore I added one ring to the chain so that the number of vibrations might turn out less and it might approach more nearly in each of it's vibrations to a second of the Primum Mobile. Now I am not giving you the rest of the text, I hope you can picture this, nine guys, one of them Grimaldi the discoverer of diffraction.
These are capable people out there counting for a 24-hour period on a pendulum and they mean one second, that's one second each way, okay? They don't mean our time. So the third example he modifies it, and he adds 0.21 inch To the second one, and gets 86,998 versus 86,400.
And finally, with the fourth example, pendulum three, old Roman feet, two at 67 and one hundredth inches, with a 20 and one half ounce brass sphere 32, 112 arcs in 3192 sidereal seconds. Now their actually looking at two different stars passing across one another, they're not going the whole day.
Based on that, he reaches a conclusion. A one second pendulum is three old Roman feet. 3 plus 27 one-hundredths inches, etc., and he's scaling, it's not exactly what he had at the top, he's doing a small correction. Then he's setting up a half second pendulum and a one-sixth second pendulum.
Six times a second. Okay, and that's actually three full arcs per second. Therefore, we used a pendulum of this sort for measuring the natural movement of weights, but in order to count its vibrations as quickly as possible, it is proper after each set of ten to raise one finger of two class cans and to be extremely attentive indeed, for greater proof to take two equal pendulums of this sort, and have two counters making their own count separately.
So it is apparent at the end of the operation where it agrees or not, I have two people doing it, okay. Next thing he does and I don't have a separate slide on this, is he takes chalk covered balls, all the same size, but with different masses of lead inside and starts dropping them from 312 feet.
You'll see how he gets 312 feet in just a moment. Getting them heavier and heavier, til he gets to a point where the distinction and the time in which they land becomes negligible. First of all, he says even for light ones versus heavy ones, Aristotle is wrong, it's not proportional.
But if you get heavy enough, you really can't tell the difference. Now we're gonna use those because those are in effect controlling for air resistance and he goes to the Towers of Bologna. This is Tower of San Petronius. You can't, it's disappointing, in this room, it's being wiped out a little, disappointing because this is my photograph, and I'm very vain about my own photographs.
This is a photograph made from the observatory at the University of Bologna, and I don't mean the current one I mean the one as of the time Richie Holt was there. Looking out over the city, you see a lot of different towers. The tallest tower is Asinelli Tower, it's 312 feet high, according to him it's actual 97.2 meters high, that's my 100 meter height.
Okay, and now I will simply describe. You can see, towers all over the place? They had a lot of towers, and by the way, that's got a slight lean on it, so it does have the virtue for dropping objects. Guess I have to read. So, the top says weights in perpendicular free fall move more and more quickly towards the end, in an increase of speed that is between numbers equally unequal numbered as wholes.
Or so as the space is traversed in certain times, are among themselves as the square of the times, and so as the space is traversed, have among themselves a duplicate proportion etc. In other words, Galileo's right and that's what he says, the whole assertion of Galileo and Baliani has been very often proven by our experiments.
Now these are the numbers are said to be equally unequal as wholes. One, three, five, seven, nine, 11, 13, 15, etc., and so if in the first quarter of an hour some weight is depleted, it's the explanation. In order to explore this in truth, Grimaldi and I prepared in advance, several clay globes of the same bulk, and dropped those of 8 ounces from different towers or house windows or little casements suited for taking measurements and we first used the towers of Polonia.
Namely the Asinelli, 312, the St. Peter, 208, the San Petronius, that's the one I showed you, 200, etc. Many, many different towers. Moreover, for discerning more precisely the time in which the drop glows arrived at the pavement, we use two very small pendulums, back to the prior reference, etc.
Each one's supposed to be one second and these are his best results. I'll just read, and then we'll look at the results. I'm gonna keep you for maybe three or four minutes, over. And so in the first experiment, when we observe from a height of ten feet from the aforesaid globe, come to the pavement in at least five vibrations of the aforesaid pendulums, five, sixths of a second.
We tested the height which a globe equal to that one passed through in ten vibrations, and found it to be 40 feet and going on through all the rest. They would do things like knowing the drop in the first second they could predict what it should be from different intermediate heights.
They did the right kind of experiment all over the place. He says these are the two best sets of results and you can see. One, three, five, seven, nine. One, three, five, seven, nine. Exact, in fact the worst thing about this, is it's exact agreement, okay? It makes one wonder, but he said, these are the two best and he got exact agreement and he's going up to 280 feet and the total time is what?
I can't quite read this on time. But you should be able to, and it's a sixth of a second. So we're running substantial periods of time over very high heights. That was the strongest evidence anybody had as of 1651, which will be a magic date for your third paper, second paper, had as of 1651, for that the one, three, five progression actually holds.
Two more slides very quickly. One of them we can look at this in retrospect and ask how good is his result. Well the way to think of this is what you want is RG, okay? The velocity acquired in a given timem, the constant of proportionality, which is the acceleration of gravity.
They didn’t think that way, they didn’t do acceleration of gravity, they did distance of fall in first second, which is half RG. Okay, and Galileo said in a letter in 1635, four cubits in the first second, 197 centimeters. Mersenne, in Harmonie Universelle, says 12 Paris feet in the first second, 394 centimeters.
Riccioli, depending on what you use as the measure of Roman feet, 15 Roman feet. If I take the 312, and the 97.2 meters, and use that as the length of his foot, I get 467. If to the contrary, I use Quarayz value in the classic paper I put on supplementary material, I get 444 centimeters, which is 10% off.
Now a comment, 10% off on all the distances makes no sense, it can't be because of resistance because that's a function of velocity squared. Can't be because of an error in measuring time because distance goes as time squared. The error has to be linear through all the cases it can only be an error in distance.
How could they be off by 10% in distance? So it was Steve Weinberg who made me appreciate this when he told me, look at the height of the 312. Get it in present units. He had the wrong units, but the right units get it to where it is halfway reasonable.
As you see, in 1659, then published in Horologium in 1673. Huygens measures to four significant figures our current number. So the Riccioli numbers, retrospectively, from 1659 which is only eight years later, are actually good to within better than 5% and within better than 5% is probably close to the accuracy with which they can observe the heights.