## Philosophy 167: Class 10 - Part 5 - Extensive Measurements of Surface Gravity: Huygens' Method.

Smith, George E. (George Edwin), 1938-

2014-11-04

These are his instructions. You'll see the date, 1686. 1686 is when Newton is busily finishing the Principia, unbeknownst to Huygens, he's busily finishing this. There's a voyage leaving from Texel. Texel is the port in North of Amsterdam, principle port coming out from the Northern part of Holland, down to the Cape of Good Hope.
They were originally intending to go on to what is now the Dutch East Indies. And what he wanted Thomas Helder to do, was to measure surface gravity all along the voyage and at every stop because Huygens wanted a complete map of the variation of surface gravity from Holland all the way down to the Cape of Good Hope in the South at the very least.
You'll see he got that. We'll look at that next semester. But here are the instructions. While ashore at the Cape of Good Hope, as well as especially in Batavia. That's East India. If the voyage goes so far. Or while the ship is lying very still, one will observe, by using the clock, how long a single pendulum must be to do each beat in a second.
That is, the length from the top end of the thread until the center of the sphere. For here, I call a simple pendulum a copper or lead, little sphere of about a thumb in diameter, that's about two inches, that is hanging on a thin thread, silk thread. The footnotes are by me and Eric Schliesser.
Eric Schliesser's the translator of this, this is all in Dutch. With regard to the motion of the clock, much depends on this experience. For a certain Frenchman, that's Jean Richer, claims to have found a location about five degrees North of the Equator, that such a pendulum was a bit shorter than here in Paris, England and Holland.
In order then to observe this perfectly, one should hang the pendulum as in the figure. I won't continue. Up with this part. Go to the next. One will make this pendulum move very slowly. Roughly just two or three thumb widths, being very careful. So we're talking about an arc length of between my two thumbs on a three foot length.
That's not very much, particularly if the sphere itself is two thumbs. Very small arc. Being very careful that the sphere no longer rotates as always occurs from the start, for through this the threat unwinds itself and becomes longer. One can impregnate it with wax. See how's he's an engineer?
He's fixing the thing. Except at near the top. Furthermore one will observe the movements of this pendulum against of the clocks ensuring that one movement accords with two movements of the pendulum of the clock. It's a half second clock. So it's going twice as fast as the one second pendulum.
And here's the part. Ensuring that one movement occurs with two movements of the pendulum clock. And that for about a half an hour, 1,800 swings of the pendulum, 3,600 swings of the clock pendulum staying in synchrony. That's a long time folks. One can thus shorten or lengthen the pendulum AB until the beats as was said accord perfectly.
Once then this has been done one shall measure off neatly with a straight stick having been shortened to this measure the true like they be etc. I'm still not done. Next one. But as the clock usually goes several seconds too quickly or too slow in 24 hours, so the movement of this single pendulum will be a bit shorter or a bit longer than the second.
Let us suppose that a clock goes one minute too slow in 24 hours, then turns 24 hours into minutes, resulting 1440 etc. Now what he's gonna do is correct the length of the pendulum for the error in the clock. Okay. If the clock loses ten seconds in the day.
You'll make an adjustment. And I'll do it this way. This is all available to you including the footnotes. This the summary. We're gonna measure the distance of fall in the first second by that formula. T here is because they didn't think of the pendulum the way we do as two arcs constitute a period.
They thought one arc was the time unit. Even though, in the case of a conical pendulum, it's a full cycle. In the case of a single pendulum they didn't think it was a full cycle, they had said. So I'm using T to represent a single arc. Length of a one second pendulum, turns out in Paris, three feet, 8.5 lines, 440 point lines.
Distance to fall 15 Paris feet, 1.1 inches, and here's the sequence. Adjust the length in synchrony with a pendulum clock for 30 minutes. Measure the length to the bob's center. Correct for the center of oscillation. Correct the length for any inaccuracy in the clock. Have to do it over a 24 hour period.
86,156 is the number in the Sidereal seconds in the day. You then do the correction. And then you get the distance of fall. Okay, nice sophisticated measurement, and how good is it? This is my own analysis. I'm not gonna go through this, you can read it on your own, but yeah, maybe I'll go through.
Suppose the accuracy of the pendulum clock could be determined within four seconds per day. That's the accuracy of the pendulum clock, not it be that accurate, but it could be determined against the stars. Suppose a second pendulum could be determined still to be one tenth of an arc in synchrony after 30 minutes.
One tenth of an arch is a pretty large discrepancy between the two. Suppose the length of the second pendulum could be determined within two tenths of a Paris line, two tenths of a twelfth of an inch. You're gonna have to use a magnifying glass to do that. Then the error in the strength of gravity is one part in 1,520.
If instead you could get down to one tenth of a Paris line, you'd have one part in 2,320. So, the claim that he was measuring to within roughly one part in a thousand, or one part in 2,000, that's a legitimate claim. That's the level of the measurement. And as I say, it became standard except people would butcher it.
Butcher it by allowing the arch to go too low. Now what might intrigue you, I'll be interested in looking at your faces when I say this, this was the method we measured surface gravity all the way down to 1950's. We got up to seven significant figures. The pendulums got better.
With more care and more corrections put in, because we knew more. But it was only with space ships, and using accelerometers in space, that we finally gave up. Using pendulum measurements for surface gravity. So, I'll show you at the end of the course, seven significant figure maps of surface gravity around the Earth, all done with pendulums, published in 1956.
So he's not only invented a method here, he invented a method that lasted a long, long time, with nothing more than refinements in the pendulum design.