The Mountain That Weighed the Planet

Eleven point six arcseconds. That is roughly the width of a human hair seen from across a room, and it is the entire signal that Britain used to weigh the Earth [S1].

The summer of 1774 found Nevil Maskelyne, the Astronomer Royal, camped on the flanks of Schiehallion, a quartzite ridge in Perthshire chosen the year before by Charles Mason after he scouted the Highlands for a mountain symmetric enough, isolated enough, and — he hoped — uniform enough to serve as a giant test mass [S1]. The Royal Society had funded the trip on the conviction that a nearby mountain ought to tug a plumb line a measurable fraction of a degree off true vertical [S1]. Newton, in the Principia, had already estimated such mountain-induced deflections would be minute, possibly hopeless to measure — the much-quoted "too small to measure" line is more a later populariser's gloss than Newton's own words [S1]. The Society wanted to try anyway, partly because a measurable pull would also refute the contemporary speculation that mountains were hollow shells [S1].

Maskelyne's apparatus was almost insultingly simple. On the north flank and the south flank he built observatories. In each he set up a zenith sector — a long telescope hung so it could swing only in the meridian — and watched stars cross overhead [S1]. The angle a star made with the local vertical was set by two things: the curvature of the Earth between the two stations, which surveyors could measure on the ground, and the local direction of gravity, which the plumb bob defined [S1]. On a featureless plain, the two would track each other. With a mountain sitting between the observers, they would not.

The plumb bobs at both stations leaned a little toward the mountain. So the apparent latitude separation of the two observatories, as read off the stars, came out larger than their surveyed separation on the ground. The numbers Maskelyne brought back: 54.6 arcseconds in the sky, 42.94 arcseconds on the ground. Difference: 11.6 arcseconds [S1]. That gap was Schiehallion pulling on string.

What you do with that gap is the part nobody warns you about. The deflection tells you the ratio of the mountain's gravitational pull to the Earth's. To get from there to a density for the planet, you also need to know the mountain — its volume, its shape, its mass. Charles Hutton, a Newcastle mathematician brought in to do the reduction, took the survey points home and disappeared for two years [S6].

His method was brute force. He sliced the mountain into a grid of vertical prisms, computed the gravitational attraction of each individual prism on the plumb bobs at the two stations, and summed thousands of contributions by hand [S6]. The output, when it finally arrived in Philosophical Transactions volume 68 in 1778, ran to roughly a hundred pages [S6]. Buried in it was the headline ratio: if Schiehallion and the Earth shared a density, the planet's pull on the plumb bob would have been 9,933 times the mountain's [S6]. Since the planet plainly pulled harder than that, the Earth had to be denser than its mountains. Hutton's number for the mean density of the Earth came out near 4.5 grams per cubic centimetre — about 1.8 times the assumed density of the Schiehallion quartzite [S1].

Modern value: 5.514 [S4]. Maskelyne and Hutton were off by under twenty per cent, using string, a star catalogue, and the geometry of triangles.

Then there is the side calculation that has become its own legend. To get the volume of a lumpy ridge from a flat survey, Hutton drew lines on his map joining points of equal altitude, then stacked the areas between those lines into a solid [S2]. Generations of popularisers have called this the first topographic contour line. The cleaner version is that he was a notable user, not the inventor. Pieter Bruinsz drew underwater contours on the Spaarne in 1584; Nicholas Cruquius mapped the Merwede with one-fathom isobaths in 1727; Philippe Buache contoured the English Channel in the 1730s and 1750s; Domenico Vandelli put contour lines on a land map of Modena and Reggio in 1746, nearly thirty years before Schiehallion; and Marcellin Ducarla theorised land contours in 1771, three years before Maskelyne pitched his tents [S3]. A 2019 paper in the Journal of Maps adds the awkward kicker: Hutton's contoured plan of Schiehallion was never actually published — only the calculations that depended on it [S2]. The technique was in the air; Hutton used it in anger on a real problem at industrial scale; the claim that he invented it is a tidier story than the record supports [S2][S3].

The bigger surprise is what happened when Henry Cavendish ran the experiment again, indoors, in 1798, with a torsion balance and two lead spheres. The standard line is that Cavendish confirmed Maskelyne. He did not. Cavendish's published figure for the mean density of the Earth was 5.48 grams per cubic centimetre, against Maskelyne's 4.5 [S4]. (Francis Baily later showed Cavendish's own arithmetic actually yielded 5.448; the published 5.48 was a small slip that happened to land closer to the modern value [S4].) A twenty per cent gap between two experiments is not a confirmation, and the discrepancy hung there unresolved for two centuries.

The resolution came in 2007. John Smallwood, in the Scottish Journal of Geology, redid Hutton's calculation using a realistic geological model of Schiehallion — not the comforting fiction of uniform quartzite, but the denser, more heterogeneous interior the mountain actually has [S4]. The new answer: 5,480 ± 250 kilograms per cubic metre, essentially the modern value and a match for Cavendish [S4]. Maskelyne's deflection measurement was sound. It was the assumption about what Schiehallion was made of that biased the Earth's density downward [S4]. The plumb-line angle had been right all along; the mountain had been quietly lying about its insides.

Which leaves the human residue. Four months in tents on a Highland ridge, in a Perthshire summer, with midges. The camp's ghillie — hired as cook and porter — was Duncan "Redhair" Robertson, who entertained the surveyors in the evenings on a fiddle [S5]. At the end-of-survey party the northern observatory was, in the Royal Society's own euphemism, accidentally burned to the ground [S5]. Robertson's fiddle went with it. Maskelyne, back in London, personally replaced the instrument; the violin he sent north is the one local tradition calls the Yellow London Lady [S5].

You can stack the ledger any way you want. Britain weighed the planet that summer to within twenty per cent on the first try, and the deflection measurement, properly reduced, is right to the modern decimal place [S1][S4]. The contour line was not invented at Schiehallion, but it was put through its first serious workout on a problem nobody else had ever needed it for [S2][S3]. Cavendish gets the textbook credit for the Earth's density; he was checking, not founding, and the check itself was off by enough that nobody noticed for two hundred years [S4]. A fiddle burned in the celebration, and the Astronomer Royal bought a new one [S5].

The lab-coat version of how big numbers enter physics — clean apparatus, controlled rooms, a single decisive measurement — is a flattering edit. The actual file is messier: a wet ridge, a long summer, a borrowed plumb bob, and a hundred-page paper from a man who spent two years adding up the gravity of imaginary prisms.