dinsdag 14 augustus 2018

Determine the shape of the Earth 200 years ago with gravity



A colleague of mine pointed me to a PhD dissertation of Petrus van Galen. He wrote this dissertation in 1830 and two copies can be found in the Tresor of the TU Delft Library. The title of this dissertation is “Disputatio Mathematica Inauguralis de Pendulo Ejusque Applicatione ad Telluris Figuram Determinandam”. Loosely translated: A Mathematical Essay about the Pendulum and its Application to Determine the Figure of the Earth. Of course, as soon I heard about this, I made an appointment with the librarian responsible for the collection in the Tresor for a viewing session. I also found a link in Google books, showing almost the complete dissertation, but I wanted to smell the old pages. Furthermore, the appendices of the dissertation were not scanned properly, so the data, that Petrus van Galen used, were not yet available to me.


A week later, I cycled to the Library (appointed as one of the 10 most beautiful libraries of the world and known very well to me due to the Vening Meinesz project) and dropped by the Tresor. In one of the copies, the pages were still joint together, as they were printed on one big sheet of paper and not yet separated by the cutting machine. I was told that this was common for books in those days. The other copy was cut, which made browsing through the dissertation a bit more easy. As you could see in the Google books link, it was almost completely written in Latin, except the preface, written in Dutch by Petrus’ friends. The dissertation started by a quote from the Roman philosopher Seneca, whom I find to have written most intriguing letters (e.g. De Brevitate vitæ). I let you do the translating. 


Because of the language it was difficult to understand the thesis, but armed with 5 years of high-school Latin and Google translate, I do understand some of the text Petrus has written. And the equations are written in the universal language of mathematics, which I am able to read. Van Galen discusses the application of a single pendulum to determine the shape of the Earth. This is a similar experiment I asked you to do in one of my previous blogsIn those days it was known that the Earth was not completely round, but had a flattening, caused by its own rotation. This flattening can be determined by observing the gravity field on the surface of the Earth. Gravity has an influence of the period of a pendulum, so by observing the period of such a pendulum, scientists were able to determine the gravity field of the Earth. The value of this gravity is different depending on the latitude of the location of the experiment. In the appendix of the dissertation, Van Galen lists several now famous scientists that did this experiment on different locations. 


People like Bouguer, Condamine, and Picard were famous geodesists, people I learned about during my study and now teach about to my students. A great Italian explorer Malaspina is listed and contributes to the data with 16 entries. Even Celsius is on the list, famous for his temperature scale, but I know him for his pioneering work on post-glacial rebound in Scandinavia. The locations where the measurements were taken are also interesting to look at. It shows that the southern hemisphere was less well covered and no observations in the oceans, which were the arguments of Prof. Vening Meinesz (about whom I have wrote several blogposts: here, here, and here) for his voyage on the K-XVIII submarine in 1934-1935. In Paris, several observations have been performed by different parties, which gives us an opportunity to determine the uncertainty of gravity observations in those days. The values there differ from 979.8-981.4 Gal, so an uncertainty of 1600 mGal, which is huge in our perspective.

All very well, but how does the shape determined by Van Galen compares with Vening Meinesz’ data, other flattening values and one of our current definition of the ellipsoid (WGS84)? To do this, I have inserted all the data from Petrus van Galen in a little Matlab script, together with the definition of an ellipsoid. The definition is as follows


I used a similar expression to fit van Galen’s data, such that we can compare the ellipsoidal shape of the gravity field of Earth. From this ellipsoid the flattening of the Earth can be determined, which van Galen listed in his dissertation, being 1 : 286.32. The current value in WGS84 is 1 : 298.257223563, whereas Newton calculated it to be 1 : 230. So, van Galen was in the same order as the current value, with only 50 data points with huge uncertainties. The data that van Galen used are plotted in the following figure in comparison with data from Vening Meinesz and other definitions of the ellipsoid.

The red data points are observations collected by van Galen and the red line is my data fit using the equation listed above as the fitting function with three variables, g_e, A, and B. Blue denotes data and fit of the Vening Meinesz’ data. Black and green are a historical (1930, black) and current (green) definitions of the Earth’s ellipsoid. The data clearly shows the improvement in uncertainty of data within the 100 years between 1830 and 1930.

That the work of Petrus van Gaalen is a great piece of science can be argued by the amount of data he collected. A nice graph in the book by Torge, “Gravimetry”, illustrated the amount of data with respect to historical periods, where it shows that in 1830 around 10-100 gravity data observations were available in the scientific community. Petrus acquired an amount of 50 data points. This means that he acquired a substantial part of the data that was available, not bad in a time without internet. Think of all the visits to the library and all the letters he must have written. Above all, it shows that the Dutch Academia was interested in gravimetric research around the beginning of the Modern Age. I am especially interested in the inter connections of scientists and explorers that did these kind of experiments. The TU Delft Library is now in contact with a translator, to see if we can find some more information in this dissertation and maybe find some hidden gems of physical geodesy history.