by Mitchell Zimmer
Lori Murray’s poster at the Statistical Society of Canada Annual
Meeting won this year’s Student Research Presentation Award making it
the fifth year in a row that award is given to a Western student.
Murray says she was happy she didn’t know about the success of the
previous four years citing, “no pressure.”
In her presentation entitled “The Construction of Edmond Halley’s 1701 Map of Magnetic Declination,” Murray outlined a solution to a mystery that is over 300 years old.
Edmond Halley was a mathematician who is best known for predicting the appearance of the comet that bears his name, but there is much more to his story. As a friend of Isaac Newton, Halley was instrumental in providing the financial resources in getting Newton’s Principia Mathematica published. This turned out be a key to Murray’s detective work.
Sailors had noticed for some time that changing positions on the Earth’s surface would lead to a shift in the readings of a magnetic compass away from true North. At the dawn of the eighteenth century, Halley set out to investigate if understanding this magnetic declination would solve the difficult problem of calculating longitude (which was a big problem before the invention of the chronograph). Halley collected data during his voyages aboard the HMS Paramore in the North and South Atlantic so that in 1701 he was able to publish a map showing lines of equal magnetic declination. Even though the map showed that magnetic declination did not follow the lines of longitude, it did help navigators to estimate their routes throughout the Atlantic.
As useful as the map was, Halley did not publish the data analysis he used to construct the map and the method had been unknown. Murray suggests that Halley constructed the arithmetical means of each data point by calculating the average longitude, latitude and the corresponding magnetic declination for a set of observations that are quite close to one another. This reduced the error of the magnetic declination. Next, Murray suggests that Halley used Newton’s Divided Difference Method to fit a polynomial to data. This formula is found in Lemma V, Book III of the Principia Mathematica.
Getting to that conclusion required Murray to digitize each line of magnetic declination by recording the position for every one degree of longitude. The digitized lines were then tested for all possible combinations of third and fourth degree polynomials using Newton’s method. She found that all lines of magnetic declination could be represented at most by a fourth degree polynomial. The fact that the calculations could reasonably be done by hand in 1701also strengthens the case. Throughout the map there is a double line that shows where the compass points true, this line is known today as the agonic line and Murray theorizes that this is the first line Halley constructed and developed the map from this baseline.
The map had a great impact on navigation when it was first published and to this day , it is still used as a reference to study change in magnetic declination.
Halley's map depicting the lines of magnetic declination. The thick double line running from the east coast of he North America through the mid-Atlantic into "The Icey Sea" is the line that shows no declination also known as the agonic line. The red circles and blue triangles, showing where Halley took his observations on his first and second voyages respectively, were added by Lori Murray to show Halley's sample size.