## Nevil Maskelyne measures the Earth's density

 In 1775 Nevil Maskelyne wrote two papers on the mountain method of measuring the Earth's density which appeared in the Philosophical Transactions of the Royal Society. We give below extracts from these papers, the first of which was his proposal to the Royal Society made in 1773, and the second of which was his report which led to the award the Copley medal of the Royal Society:-

### A Proposal for Measuring the Attraction of Some Hill in This Kingdom by Astronomical Observations

If the attraction of gravity be exerted, as Sir Isaac Newton supposes, not only between the large bodies of the universe, but between the minutest particles of which these bodies are composed, or into which the mind can imagine them to be divided, acting universally according to that law by which the force which carries on the celestial motions is regulated; namely, that the accelerative force of each particle of matter, towards every other particle, decreases as the squares of the distances increase; it will necessarily follow, that every hill must, by its attraction, alter the direction of gravitation in heavy bodies in its neighbourhood, from what it would have been from the attraction of the earth alone, considered as bounded by a smooth and even surface. For, as the tendency of heavy bodies downwards, perpendicular to the earth's surface, is owing to the combined attraction of all the parts of the earth on it, so a neighbouring mountain ought, though in a far less degree, to attract the heavy body towards its centre of attraction, which cannot be placed far from the middle of the mountain. Hence the plumb-line of a quadrant, or any other astronomical instrument, must be deflected from its proper situation, by a small quantity towards the mountain; and the apparent altitudes of the stars, taken with the instrument, will be altered accordingly.

It will easily he acknowledged, that to find a sensible attraction of any hill, from undoubted experiment, would be a matter of no small curiosity, would greatly illustrate the general theory of gravity, and would make the universal gravitation of matter as it were palpable, to every person, and fit to convince those who will yield their assent to nothing but downright experiment. Nor would its uses end here, as it would serve to give us a better idea of the total mass of the earth, and the proportional density of the matter near the surface, compared with the mean density of the whole earth. The result of such an uncommon experiment, which I should hope would prove successful, would doubtless do honour to the nation where it was made, and the society which executed it.

Sir Isaac Newton gives us the first hint of such an attempt, in his popular Treatise of the System of the World, where he remarks, "That a mountain of a hemispherical figure, 3 miles high and 6 broad, will not, by its attraction, draw the plumb-line 2 minutes out of the perpendicular." It will appear, by a very easy calculation, that such a mountain would attract the plumb-line 1'18" from the perpendicular.

But the first attempt of this kind was made by the French academicians, who measured 3 degrees of the meridian near Quito in Peru, and who endeavoured to find the effect of the attraction of Chimboraço, a mountain in that neighbourhood, which is elevated near 4 miles above the sea, though only about 2 miles above the general level of the province of Quito. By their observations of the altitudes of fixed stars, taken with a quadrant of 21/2 feet radius, they found the quantity of 8" in favour of the attraction of the mountain, by a mean of their observations. This indeed was much less than they expected; but then it is to he considered, that their instrument was too small and imperfect for the purpose; and that they themselves were subject to great inconveniencies, being sheltered from the wind and weather by nothing but a common tent, and placed so high up the mountain as the boundary where the snow begins to lie unmelted all the year round. And indeed their observations, doubtless owing to these causes of error, differ greatly from each other, and are therefore insufficient to prove the reality of an attraction of the mountain Chimboraço, though the general result from them is in favour of it ...

### An Account of Observations Made on the Mountain Schehallien for Finding Its Attraction

...

Perthshire afforded a remarkable hill, nearly in the centre of Scotland, of sufficient height, tolerably detached from other hills, and considerably larger from east to west than from north to south, called by the people of the low country Maiden-pap, but by the neighbouring inhabitants, Schehallien; which, I have since been informed, signifies in the Erse language, constant storm; a name well adapted to the appearance which it so frequently exhibits to those who live near it, by the clouds and mists which usually crown its summit. It had also the advantage, by its steepness, of having but a small base from north to south; which circumstance, at the same time that it increases the effect of attraction, brings the two stations on the north and south sides of the hill, at which the sum of the two contrary attractions is to be found by the, experiment, nearer together; so that the necessary allowance of the number of seconds, for the difference of latitude due to the measured horizontal distance of the two stations, in the direction of the meridian, would be very small, and consequently not subject to sensible error from any probable uncertainty of the length of a degree of latitude in this parallel. For these reasons the mountain Schehallien was chosen, in preference to all others, for the scene of the intended operations, and it was concluded to make the experiment in the summer of the year 1774 ...

The quantity of attraction of the hill, the grand point to be determined, is measured by the deviation of the plumb-line from the perpendicular, occasioned by the attraction of the hill, or by the angle contained between the actual perpendicular and that which would have obtained if the hill had been away. The meridian zenith distances of fixed stars, near the zenith, taken with a zenith sector, being of all observations hitherto devised capable of the greatest accuracy, ought by all means to be made use of on this occasion: and it is evident, that the zenith instrument should be placed directly to the north or south of the centre of the hill, or nearly so. In observations taken in this manner, the zenith distances of the stars, or the apparent latitude of the station, will be found as they are affected by the attraction of the hill. If then we could by any means know what the zenith distances of the same stars; or what the latitude of the place would have been, if the hill had been away, we should be able to decide on the effect of attraction. This will be found, by repeating the observations of the stars at the east or west end of the hill, where the attraction of the hill, acting in the direction of the prime vertical, has no effect on the plumb-line in the direction of the meridian, nor consequently on the apparent zenith distances of the stars; the differences of the zenith distances of the stars taken on the north or south side of the hill, and those observed at the east or west end of it, after allowing for the difference of latitude answering to the distance of the parallels of latitude passing through the two stations, will show the quantity of the attraction at the north or south station. But the experiment may he made to more advantage on a hill like Schehallien, which is steep both on the north and south sides, by making the two observations of the stars on both sides; for the plumb-line being attracted contrary ways at the two stations, the apparent zenith distances of stars will be affected contrary ways; those which were increased at the one station being diminished at the other, and consequently their difference will be affected by the. sum of the two contrary attractions of the hill. On the south side of the hill, the plumb-line being carried northward at its lower extremity, will occasion the apparent zenith, which is in the direction of the plumb-line, continued backwards, to be carried southward, and consequently to approach the equator; and, therefore, the latitude of the place will appear too small by the quantity of the attraction; the distance of the equator from the zenith being equal to the latitude of the place. The contrary happens on the north side of the hill; the lower extremity of the plumb-line, being there carried southward, will occasion the apparent zenith to be carried northward, or from the equator; and the latitude of the place will appear too great by the quantity of the attraction. Thus the less latitude appearing too small by the attraction on the south side, and the greater latitude appearing too great by the attraction on the north side, the difference of the latitudes will appear too great by the sum of the two contrary attractions; if, therefore, there is an attraction of the hill, the difference of latitude by the celestial observations ought to come out greater than what answers to the distance of the two stations measured trigonometrically, according to the length of a degree of latitude in that parallel, and the observed difference of latitude subtracted from the difference of latitude inferred from the terrestrial operations, will give the sum of the two contrary attractions of the hill. To ascertain the distance between the parallels of latitude passing through the two stations on contrary sides of the hill, a base line must be measured in some level spot near the hill, and connected with the two stations by a chain of triangles, the direction of whose sides, with respect to the meridian, should be settled by astronomical observations.

If it be required, as it ought to be, not only to know the attraction of the hill, but also from it the proportion of the density of the matter of the hill to the mean density of the earth; then a survey must be made of the hill, to ascertain its dimensions and figure, from which a calculation may be made, how much the hill ought to attract, if its density was equal to the mean density of the earth; it is evident, that the proportion of the actual attraction of the hill, to that computed in this manner, will be the proportion of the density of the hill to the mean density of the earth.

Thus there were three principal operations requisite to be formed.

1. To find by celestial observations the apparent difference of latitude between the two stations, chosen on the north and south sides of the hill.

2. To find the distance between the parallels of latitude.

3. To determine the figure and dimensions of the hill ...

The difference of latitude found by the astronomical observations, comes out greater than the difference of latitude answering to the distance of the parallels, the former being 54".6, the latter only 42".94. The difference 11".6 is to be attributed to the sum of the 2 contrary attractions of the hill.

The attraction of the hill, computed in a rough manner, on supposition of its density being equal to the mean density of the earth, and the force of attraction being inversely as the square of the distances, comes out about double this. Whence it should follow, that the density of the hill is about half the mean density of the earth. But this point cannot be properly settled till the figure and dimensions of the hill have been calculated from the survey, and thence the attraction of the hill, found from the calculation of several parts of it, into which it is to be divided, which will be a work of much time and labour; the result of which, will he communicated at some future opportunity.

Having thus come to a happy end of this experiment, we may now consider several consequences flowing from it, tending to illustrate some important questions in natural philosophy.

1. It appears from this experiment, that the mountain Schehallien exerts a sensible attraction; therefore, from the rules of philosophising, we are to conclude, that every mountain, and indeed every particle of the earth, is endued with the same property, in proportion to its quantity of matter.

2. The law of the variation of this force, in the inverse ratio of the squares of the distances, as laid down by Sir Isaac Newton, is also confirmed by this experiment. For, if the force of attraction of the hill has been only to that of the earth, as the matter in the hill to that of the earth, and had not been greatly increased by the near approach to its centre, the attraction must have been wholly insensible. But now, by only supposing the mean density of the earth to be double that of the hill, which seems very probable from other considerations, the attraction of the hill will be reconciled to the general law of the variation of attraction in the inverse duplicate ratio of the distances, as deduced by Sir Isaac Newton from the comparison of the motion of the heavenly bodies with the force of gravity at the surface of the earth; and the analogy of nature will be preserved.

3. We may now, therefore, be allowed to admit this law; and to acknowledge, that the mean density of the earth is at least double of that at the surface, and consequently, that the density of the internal parts of the earth is much greater than near the surface. Hence also, the whole quantity of matter in the earth will be at least as great again as if it had been all composed of matter of the same density with that at the surface; or will be-about 4 or 5 times as great as if it were all composed of water. The idea thus afforded us, from this experiment, of the great density of the internal parts of the earth, is totally contrary to the hypothesis of some naturalists, who suppose the earth to be only a great hollow shell of matter; supporting itself from the property of an arch, with an immense vacuity in the midst of it. But were that the case, the attraction of mountains, and even smaller inequalities in the earth's surface, would be very great, contrary to experiment, and would affect the measures of the degrees of the meridian much more than we find they do; and the variation of gravity in different latitudes, in going from the equator to the poles, as found by pendulums, would not he near so regular as it has been found by experiment to be.

4. The density of the superficial parts of the earth, being, however, sufficient to produce sensible deflections in the plumb-lines of astronomical instruments, will thus cause apparent inequalities in the mensurations of degrees in the meridian; and, therefore, it becomes a matter of great importance to choose those places for measuring degrees, where the irregular attractions of the elevated parts may he small, or in some measure compensate one another; or else it will he necessary to make allowance for their effects, which cannot but be a work of great difficulty, and perhaps liable to great uncertainty.

After all, it is to be wished, that other experiments of the like kinds with this were made in various places, attended with different circumstances. We seldom acquire full satisfaction from a single experiment on any subject. Some may doubt, whether the density of the matter near the surface of the earth may not be subject to considerable variation; though perhaps, taking large masses together, the density may be more uniform than is commonly imagined, except in hills that have been volcanoes. The mountain Schehallien, however, bears not any appearance of having ever been in that state; it being extremely solid and dense, and seemingly composed of an entire rock. New observations on the attraction of other hills, would tend to procure us satisfaction in these points.

JOC/EFR March 2006

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