Boston: Trigonometry has troubled many a school student across generations (sin, cos what?), but what math teachers often fail to tell their classes is the very real applications it has.
Back in the 19th century, long before satellites emerged as the go-to tool for cartographers, it was the principles of trigonometry that helped India’s colonial rulers draw up the first accurate map of the subcontinent.
Starting from contemporary India’s southern end, up till Mount Everest, Kashmir and present-day Pakistan, it was a mammoth exercise that took years and several torchbearers to complete.
It was in the 18th century that the British East India Company started expanding their foothold in the Indian subcontinent.
With rapid territorial gains came new geographical information, and a need was felt to fit this data into a recognisable cartographic image: The British realised that to establish political administration and military control, they needed maps.
The first systematic survey of the region resulted in the compilation of the Bengal Atlas, which combined topographical measurements with astronomical observations.
However, the imprecision in the methods used was increasingly apparent. The astronomical observations were often inconsistent, and the topographical measurements, carried out through “route surveys” performed on foot, were found to be inaccurate.
The British soon realised that to gain complete geographical knowledge of a vast uncharted territory, this imprecision was unacceptable.
The Great Trigonometrical Survey of India
Cue in British army officer William Lambton, a geographer and geodesist who had used his expertise in celestial navigation to guide the British troops in the right direction during the 1799 siege of Srirangapatnam, thereby aiding in the defeat of the Mysore Empire of Tipu Sultan.
Lambton understood the necessity of having scientifically-accurate maps of the newly-conquered territory, and he found an answer in trigonometry, as Clements R. Markham, who served as president of London’s Royal Geographical Society in the 19th century, explains in his book A Memoir on the Indian Surveys.
Triangulation as a way of measuring distances is believed to be the brainchild of a Dutch professor named Willebrord Snell. Seeking to apply the principle to India, in 1802, Lambton established a “baseline” at St Thomas Mount in what was then Madras.
From the two end points of this baseline, whose length was measured with great care, line-of-sight observations were made to a third point.
By observing the interior angles of the triangle made by these three points, Lambton was able to ascertain the length of the other two sides of the triangle, and thereby deduce the exact position of the third point.
These two newly-calculated sides then became the baselines for additional triangles. A network of such triangles was constructed with all points defined with respect to one another, anchored to one precise measurement.
Lambton envisioned that these triangles would be the key that unlocked the mysterious geography of the Indian peninsula. He wanted the Malabar coast in the west to be connected to the coast of Coromandel on the east with “an uninterrupted series of triangles”, continuing that series to “an almost unlimited extent in every other direction”, Lambton wrote in his book, An Account of a Method for Extending a Geographical Survey across the Peninsula of India.
“[A]fter surveys of different districts are made in the usual mode, they can be combined into one general map,” he added.
By 1818, Lambton’s undertaking, then still underway, was officially named the Great Trigonometrical Survey of India.
The half-tonne tool
To carry out his measurements, Lambton had, among other instruments, a 100-foot chain to measure the baselines, and a half-tonne tool called “The Great Theodolite” that was employed to measure horizontal and vertical angles.
The Great Theodolite required a dozen men to carry it. British historian John Keay, in his book The Great Arc (2000), wrote that the Great Theodolite was one of “probably only two or three instruments in the world” sophisticated enough to serve Lambton’s purpose.
When making line-of-sight observations proved difficult in the tree-choked Kaveri delta, Lambton resorted to scaling the gopurams of Tamil temples to rise above the thickets. Hoisting the half-tonne theodolite on top of the gopurams was, needless to say, no mean task.
In 1808, when the expedition was in Tanjore [now Thanjavur], Lambton attempted to haul the Great Theodolite atop the “vimana” of the Brihadeesvara Temple.
A guy rope [tensioned wire] used to keep the theodolite clear of the pyramidal tower failed, and the instrument crashed against the temple’s tower and suffered damage that would temporarily take it out of commission. Lambton himself had to restore the Great Theodolite, making sure that it met his exacting standards.
The chain, meanwhile, had to be used with supporting gear that kept it perfectly horizontal and at a consistent tension. However, the sun’s heat presented the problem of the chain expanding, which could have resulted in inaccuracies in the resulting maps.
These problems were solved, Markham observed in his book, when a set of “compensation bars” replaced the chains. The compensation bars were made of two metals — brass and iron—and constructed in a way that their different rates of expansion would cancel each other out and keep the bar itself in the same shape.
In 1830, when the newly-ordered compensation bars had to be calibrated, the surveyors found just the flat surface needed to carry out their experiments at Lord’s Cricket Ground.
The Great Arc
The central column on which the Great Trigonometrical Survey of India depended was a chain of triangles that ran north from Cape Comorin (now Kanyakumari), roughly following the 78-degree meridian. This series of triangles became known as the Great Arc of the Meridian.
Around 250 BC, the Greek astronomer Eratosthenes became the first person to use arc measurements for scientific reckoning.
He knew that the sun, at noon of the summer solstice, was directly overhead in the city of Syene, but not in Alexandria.
He calculated the angle of the sun’s rays in Alexandria by measuring the length of the shadow cast by a vertical rod. Combining this information with his knowledge of the distance between Syene and Alexandria, Eratosthenes calculated the size of the earth.
In the 1730s, French scientists had made two arc measurements: One at the equator in what is now Ecuador, and the other in the Arctic Circle in Lapland, Finland.
These expeditions demonstrated that the Earth was more curved at the equator, and flatter at the poles, thus furthering human understanding of the Earth’s shape.
Lambton wanted to know whether this distortion in curvature was of a consistent form. He realised that the Great Arc, being the first arc measurement in the tropical latitudes, could help answer this question.
A country mapped
Lambton’s mammoth project outlasted him: When he died in 1823, his assistant took over the reins.
The assistant then was replaced by Andrew Scott Waugh, a British army officer who also served as India’s surveyor general, in 1843. By then, the survey had begun to cover not just the length and the breadth of the country, but also its heights.
In 1856, Waugh announced the discovery of what he believed to be the highest peak in the Himalayas. He decided to name it after his predecessor, the man who spearheaded the survey from 1823 to 1843: George Everest.
You guessed it: The peak is what we know today as Mount Everest, the world’s highest mountain.
By mapping the entire Indian subcontinent with scientific precision, the survey made possible the development of roads, railways and telegraphs.
By producing new values for the curvature of the Earth, it significantly advanced knowledge of the exact shape of the planet.
Markham observed in A Memoir on The Indian Surveys that the Great Arc, the backbone on which the survey rested, was “one of the most stupendous works in the whole history of science”.
Abhinav Srinivasan lives in Boston and works in the tech industry. You can follow him on Twitter @thexfactorial.