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Colonel's rules and his own, though these differences are very small, is the design of this paper ; in the course of which it appears that the two principal causes of difference arise from the expansion of quicklilver and the expansion of air. The difference arising from the former Sir George Mhews can seldom amount to more than about 5 feet in a height of 11,000. In their equations for the expansion of air, the difference is indeed greater, and may be 4 feet in 1000, if the mean height of the two barometers be 27 inches, and the thermometer stand at 52o. The error increases as the difference between the heights of the two barometers and the height of the thermometer increases. Sir George subjoins fome reasons for suspecting the accuracy of those observations, which seem to require an equation that depends on the latitude of the place.

MATHEMATICA L. Article 33. An Account of the Calculations made from the Survey

and Measures taken at Sihehallien, in order to ascertain the mean Density of the Earth. By Charles Hutton, Esq; F. R.S.

The Newtonian philosophy supposes that attraction is exerted not only between the great bodies which compose the universe, but also between the moft minute particles of matter which those bodies consist of: hence it is evident, fuppofing this doctrine to be true, that the plumb-line of a quadrant, or any other astronomical instrument, situated on the side of a very high hill, or in its neighbourhood, must be attracted from its proper perpendicular direction by the matter in fuch hill; and of consequence, the meridian altitude of any star, observed with that inftrument, will be different from what it ought otherwise to be : and, moreover, if the meridian altitude of the fame ftar be observed both on the north and south sides of the hill, the attractions in these two cases being different ways, the difference of the two meridian altitudes, when corrected for the difference of the two geographical situations of the instrument, will be proportional to the sum of the two attractions.

It was on this principle that Dr. Maskelyne, Astronomer Royal, about the year 1770 or 1771, proposed to the Royal Society, to determine, by experiment, the truth or falsehood of the Newtonian syitem of gravitation. The thought was not new, as the Doctor himself remarks, in his proposals, but was attempted once before by the mathematicians who went from France, about the year 1736, to measure a degree of the meridian, on or very near to the equator : the manner, however, in which the experiment was then conducted, was by no means such as to give satisfaction to philosophers in general; and if it had, the importance and delicacy of the experiment is so great as still to merit many repeated trials; for as the Author of the paper

now

now under consideration justly observes, ' a frequent repetition of the fame experiment, and a coincidence of the results, afford that firm dependance on the conclufions, and satisfaction to the mind, which can scarcely ever be had from a single trial, however carefully it may be executed.' The result of the experiment, as well as the manner of conducting it, is related at large by the Doctor himself in the Philofophical Transactions for 1775, and an account of it was given in our Review for June 1776. From whence it appears that the plumb-line of the instrument was deflected from its true perpendicular direction, by the attraction of the mountain, by an angle of about 51 se. conds: the sum of the two deviations being i". 6; and which establishes the truth of the Newtonian philosophy on the solid foundation of experiment.

It remained still to determine, from this most curious experiment, the ratio of the mean density of the hill to that of the earth, and from hence, and the known matter of which the hill consists, that of the latter to common water, or any other known fubftance. This is the purport of the paper before us, which takes up one hundred pages of the Transactions : for as the businefs was in its nature entirely new, it laid Mr. Hutton under the necesity of inventing, and, of course, defcribing at length, the several modes of computation which he has made vse of, and also of giving a synopsis of the measures which were taken of the several lines and angles, that any person, who thinks proper, may satisfy himself of the truth of the computations here delivered.

It appears that two principal bases were measured, befide other fhorter lines, one on the south, and the other on the north-west fides of the mountain. From these two bases, and the several angles which were also measured, both vertical and horizontal, from their several extremities to different parts of the summit and base of the mountain, as well as different points on its surface, the plan of it, as well as the figure of a prodigious multitude of sectio were computed ; and from thence also the figure of the bill was constructed, on a very large scale, upon paper.

Notwithstanding this stupendous piece of computation was thus effected, one, not less arduous, appeared behind, which was to apply the foregoing calculations and constructions to the determination of the effect of the attraction of the mountain in the direction of the meridian: and here it soon occurred to the ingenious Computer, that the best method would be to divide the plan into a great number of small parts, which might be conlidered as the bases of so many small columns, or pillars of matter into which the hill and the adjacent ground was divided by vertical planes, forming an imaginary groupe of vertical coa lumns, something like a set of basaltine pillars, or like the cells in a piece of honey-comb; then to compute the attraction of each pillar separately in the direction of the meridian ; and, lastly, to take the sum of all these computed effects for the whole attraction of the matter in the hill. It is obvious that the attraction of any one of these pillars, on a body in a given place, may be easily computed, and that in any direction, because of the smallness and given position of its base : for on account of its smallness all the matter in the pillar may be supposed to be collected into its axis or vertical line, erected on the middle of its base, the length of which axis, as the mean altitude of the pillar is to be estimated from the altitude of the points in the plan which fall within and near the base of the pillar : then, having given the altitude of this axis, together with the position of the base, and the matter supposed to be contained in the pillar, and collected into the axis, a theorem is easily derived, by which the effect of its attraction may be computed. But to retain the proper degree of accuracy in this computation, it is evident that the plan must be divided into a very great number of parts indeed, to have the pillars fufficiently small to admit of this mode of computation, not less than a thousand for each observatory, or two thousand in the whole, forming the bases of as many such pillars of matter as have been described above; which, if the attractions of every one had been separately computed, must evidently have been a work of such labour as would have discouraged every person from undertaking it; but which must nevertheless have been the case if our Author had not luckily hit upon a method of dividing his miatter into columns, so as to abridge the computations in a most remarkable manner ; but which, as well on account of the want of proper diagrams, as the great length of the process by means of which it is derived, cannot pollibly be pointed out here : suffice it to say, that the result of this long and intricate calculation was, ihat the effect of the attraction of the matter in the mountain and adjacent hills, at the southern observatory, was to the effect of the same attraction at the northern one, as 69967 to 88644, or as 7 to 9 very near. This difference, Mr. Hutton shews, is to be attributed principally to the effect of the hills which lie on the south side of the mountain Schehallien, and which are not only larger, but also nearer to it than those which are on its north side.

Mr. Hutton next proceeds to compare this attraction with that of the whole earth, and finds, taking a mean of all the measures which have been given for the length of a degree of one of its great circles, that the whole attraction of the earth is to the sum of the two contrary attractions of the hill as 87522720 is to 88111; that is, as 9933 to i, very near; obferving, that this conclusion is founded on the supposition that the denfity of the matter in the hill is equal to the mean denfity of all the matter in the earth. But the Astronomer Royal found, by his observations, that the sum of the deviations of the plumb-line produced by the two contrary attractions of the mountain was 11". 6: from which circumstance it may be inferred, that the attraction of the earth is actually to the sum of the two contrary attractions of the hill, as radius to the tangent of 11". 6, nearly; that is, as i to .000056239, or as 17781 to 1. Or, after allowing for the centrifugal force arising from the rotation of the earth about its axis, as 17804 to I nearly. Having thus obtained the ratio which actually exists between the attraction of the whole earth and that of the mountain, resulting from the observations, and also the ratio of the same things arising from the computation, on the supposition of an equal density; the Computer proceeds to compare these two ratios together, and by that means determines that the mean denfity of the whole earth is to that of the mountain as 17804 to 9933, or as 9 to 5 nearly.

On reviewing the several circumstances which attended this experiment, and the computations made from it, Mr. Hutton concludes that this proportion must be very near the truth : probably within a fiftieth, if not the one hundredth part of its true quantity. But another question yet remains to be determined, namely, what is the proportion between the density of the matter in the hill, and that of some known substance ; for example, water, stone, or fome one of the metals? In this point, the Author observes, any considerable degree of accuracy can only be obtained by a close examination of the internal structure of the mountain : and he chinks that the easiest method of doing this would be by boring holes, in several parts of it, to a sufficient depth, in the saine manner that is done in searching for coal-mines, and then taking a mean of the denfities of the several strata which the tool passes through, as also of the quantities of matter in each ftratum. Bat as this has pot been done, we must rest satisfied with the estimate arising from the report of the external view of the hill, which, to all appearance, consists of an entire mass of solid rock : Mr. Huta ton thinks, therefore, that he will not greatly err by afiuming the density of the bill equal to that of common stone, which is not much different from the mean density of the whole matter, near the earth's surface, to such depths as have hitherto been explored, either by digging or boring. Now the density of common stone is to that of rain-water as 21 to 1; which being

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compounded with the proportion of 9 to 5, found above, gives 41 to 1 for the ratio of the mean density of the whole earth to that of rain-water. Sir Isaac Newton thought it probable that this proportion might be about 5 or 6 to 1: so much justness was there even in the surmises of this wonderful man !

Mr. Hutton proceeds to observe, that as the mean denfity of the earth is bout twice the density of the matter near the surface, there must be somewhere, towards the more central parts, great quantities of metals, or other very dense substances, to counterbalance the lighter materials, nearer to the surface, and produce such a confiderable mean density. He then goes on, having the ratio of the mean density of the earth to that of water, and the relative densities of the planets one to another, determined from phyfical considerations, to find their densities relative to rain water, which he makes as follows: Water

Mars

34 The Sun

Moon

311
Mercury

97
Jupiter

11t
Venus

513 Saturn The Earth

43 Mr. Hutton concludes his paper with pointing out some particulars which may tend to render the experiment more complete and accurate if it should ever be repeated. Article 41. A Method of finding, by the Help of Sir Isaac New

ton's binomial Theorem, a near Value of the very slowly-converging infinite Series ****+*+*+&c. when x is very nearly

2 3 4 equal to 1. By Francis Maseres, Esq; F.R.S. Cursicor Baron of the Exchequer. If A, B, C, D, &c. be put for the numerical co-efficients of X and its powers in the above series, it is manifest, A being equal 1, B=, C=}, D=, &c. that B will be equal to 1A, C=B, D={C, and so on; and consequently, by fubftituting these quantities for their equals in the original series, it will become x+? Ax?+;Bx}+{Cx++&c. where it may be observed that the fractional, or numerical part of the co-efficient of each term, after the first, is derived by adding 1 both to the numerator and denominator. It will also be found, by Sir Isaac Newton's theorem, that the binomial 1

* is equal to the series it-Ax + mtn Br+m+an

Cx3 +

m+39 Dxit &c. 2n

3n where the capitals 4, B, C, D, &c. ftand for the fractional part of the co-efficient of the preceding term ; and it is obvious that these fractional parts are constituted by adding n both to the numerator and denominator of the co-efficient of the term

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