In certain circles recent work by Dr. Daniel Mansfield, a mathematics professor at the University of South Wales – Sydney, is creating a buzz as he is once again looking at a Babylonian tablet of geometry. This one, Tablet Si.427, is located at a museum in Istanbul, and may be addressing the issue of “Pythagorean Triples” as used by the Babylonians a millennium before the time of Pythagoras. I suspect that some of you have started to wonder about continuing to read this, but bear with me, I’ll try to make it interesting. You see, this has something to do with how we map things, including mapping things accurately that are far away. I’ll not get into the math (very much), but will tell you the story about what the Anglo-Indian surveyors were up against in their dealings with the Kingdom of Nepal back in the mid-1800’s.
Pythagoras (570-495 BC) was, of course, the Greek philosopher generally credited with the discovery, or at least the first proof, of the Pythagorean Theorem. It seems likely that the mathematical concept we call Pythagorean “Triples” was in play a good 1000 years (or more) before Pythagoras and may have been developed independently in a few different places in the world (another story for another time…).
The theorem is relatively straightforward: a^2 + b^2 = c^2, where a = perpendicular, b = baseline, and c = hypotenuse of a right triangle. The simplest solution to the equation is found by the use of the numbers 3, 4, and 5 respectively for a, b, and c. These numbers work because 3^2= 9 and 4^2= 16. 9 + 16 = 25, which is, of course, 5^2. This is known as a Pythagorean “triple”. There are many more (the next one is 5, 12, 13; the one after that is 8, 15, 17). They have been used for millennia to make accurate land surveys.
It works like this: Use a compass to lay out a N-S line. Lay out a chain (or a rod) of 4 unit lengths (baseline) along your N-S line. At the N end of the line, lay out the 3 unit length chain sideways, making it as perpendicular as you can by eye. Lay out your 5 unit length chain (hypotenuse) starting at the S end of your baseline and make it intersect the open end of your 3 unit perpendicular. You’ve just created a right triangle on the ground; moreover you’ve laid out the perpendicular on an exact E-W line. Congratulations, you’re now in the surveying business.
It appears that the Babylonian (and other early) mathematicians were primarily interested in Pythagorean Triples with respect to the lengths of each side of the triangle. Pythagoras and the other Greek mathematicians went one better by considering the angles between. Such was the birth of that branch of geometry known as Trigonometry. Once you brought angles into play, all kinds of interesting things became possible…
The first great Trigonometric Survey occurred in Europe starting late in the 17th Century and was spearheaded by the Italian-turned-French astronomer and mathematician Giovanni Domenico Cassini (Jean-Dominique). His astronomical and mathematical skills were needed in order to conduct a meridional arc survey, which addresses the question: “Does the meridian line maintain a constant rate of curvature?” (i.e., is the earth a perfect sphere?) as you move from the equator towards the poles (spoiler alert: the Earth is not a perfect sphere). Of course Cassini first had to accurately locate the meridian line which was done by watching the moons of Jupiter revolve around the planet and keeping time with a pendulum clock (which was the most accurate clock of the time). Four generations of the family were engaged in the creation of “La Carte de Cassini”, the great map of France. So the Cassini family developed the tools, techniques and analyses necessary to do very careful, very accurate mapping over large distances. However it was the British who “went big” with the idea…
It is known to students of surveying history simply as the “GTS”, the Great Trigonometric Survey of India. The project was to map the Indian subcontinent in exacting detail. How exacting? We’ll get to that in a moment. It started in 1802 just west of Madras (modern Chennai). Major William Lambton was first in charge, followed by Sir George Everest (hmm?…), Andrew Waugh, and finally James Walker, who oversaw the completion of the great survey in 1871. When you look at a map of the survey itself, you are struck by the resemblance to scaffolding around a building:
Look carefully, note that the scaffolding is composed of…triangles. Triangulation. Trigonometry. Look very carefully along the near-horizontal NE border of India with Nepal. You’ll see what appears to be a series of overlapping triangles, but they converge on points not on a line as in the rest of the scaffolding. The tops (points) of these triangles are the great peaks of the Himalaya. Throughout the time of the GTS, the English were denied access to the Kingdom of Nepal. No matter, the surveyors could see the Himalayan peaks from Indian soil (over 100 miles distant), and could triangulate both the location (horizontal triangles) and elevation (vertical triangles) without entering Nepal. Superintendent Andrew Waugh sent surveyor James Nicolson to the borderlands in 1849 to measure Mt. Everest. In 1852, the brilliant Indian mathematician Radhanath Sikdar, using the survey data provided by Nicolson, performed the first calculations on the height of Mt. Everest. His determination: 8839 meters, 29,000 feet above sea level. The best current measurement we have with modern surveying technology: 29,032’. How about that?…
