now under confideration justly observes, ' a frequent repetition of the fame experiment, and a coincidence of the results, afford that firm dependance on the conclusions, 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 Philosophical 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 5! re. conds: the sum of the two deviations being u". 6; and which establishes the truth of the Newtonian philosophy on the solid foundation of experiment. It remained ftili 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 substance. This is the purport of the paper before us, which takes up one hundred pages of the Transactions : for as the business was in its nature entirely new, it laid Mr. Hutton under the necessity of inventing, and, of course, describing at length, the several modes of computation which he has made use 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 sarisfy himself of the truth of the computations here delivered. It appears that iwo principal bases were measured, beside other shorter lines, one on the south, and the other on the north-west Gides of the mountain. From there iwo 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 its sections were computed; and from thence alio the figure of the hill 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 heft method would be to divide the plan into a great number of small parts, which might be conlidered as the balts of so many small columns, or pillars of matter into which the hill and the adjacent ground was divided D4 by by vertical planes, forming an imaginary groupe of vertical columns, something like a fet of balaltine 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, Jastly, 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, muft 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 matter into columns, so as to abridge the computations in a moft 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, that 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 ear is to the sum of the two contrary attractions of the hill as 87522720 is to 88115; that is, as 9933 to 1, 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 density 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 1". 6, nearly; that is, as i to .oc0056239, 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, refulting from the observations, and also the ratio of the fame things arifing from the computation, on the suppolition 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 carth 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 some 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 Itructure of the mountain : and he thinks that the easiest me, thod of doing this would be by boring holes, in several parts of it, to a sufficient depth, in the fame manner that is done in searching for coal-mines, and then taking a mean of the denfities of the several strata which the tool pailes through, as also of the quantities of matter in each stratum, But as this has not 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. Hute ton thinks, therefore, that he will not greatly err by assuming the density of the hill 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 fommon Rone is to that of rain-water as 21 to 1; which being compounded I 1 13 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 furmises of this wonderful man ! Mr. Hutton proceeds to observe, that as the mean density of the earth is about 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 denle substances, to counterbalance the lighter materials, nearer to the surface, and produce such a considerable 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 34 311 Jupiter lz'I Venus 51% Saturn The Earth 4 Ž 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 Newton's binomial Theorem, a near Value of the very slowly-converging x3, x4 infinite Series x+*+-+*+&c. when. * is very nearly 3 equal to 1. By Francis Maseres, Esq; F. R. S. Curfitor 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=1, C=;,.D=i, &c. that B will be equal to } ], C=B, D=1C, and so on; and consequently, by substituting these quantities for their equals in the original series, it will become x+{ Ax+iBx: +{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 n is equal to the m+n mtan Bx+ feries it 4x+ m+31 Dx++ &c. Cx} + 31 where the capitals A, 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 immediately XX 2 4 n 2 n m it'; 2n Bx+ 2 n 3 n 4 n immediately preceding. Hence if n be assumed very great in refpect of m: for example if m'be taken =1, and n = one billion, or the square of one million, it is evident that the series Ax m+ Bx+m+2n Cx' + &c. will be very nearly + 31 equal to the series + Axt 2n.Cx3+ &c. that is, 3n to it". Ax+iBx' +4Cx3+ &c. by abbreviation. Or, by re'ftoring the values of m, A, B, C, &c. to 1+x+*+ *+ *++ &c. Hence, therefore, it is evident that the binomial 27 (1997) is very nearly equal to the series 1+; *+ ***+ ***+ &c. and multiplying by n, nx1==3=nt*+-++*+ &c. and subtracting in, nxi =x+ +*++**+ &c. n teit &c. n being 1 with 3 + any very great number of cyphers annexed, and x any number whatsoever, not much exceeding unity. Article 42. A Method of extending Cardan's Rule for resolving one Case of a cubic Equation of this Form, x*--qx=r, to the other Case of the same Equation, which it is not naturally fitted to solve, and which is therefore often called the irreducible Cafe. By Francis Maseres, Esq; F.R.S. Cursitor Baron of the Exchequer. It is well known to all persons conversant with algebra, that Cardan's rule for solving the cubic equation, **-qx=r, is only suited to resolve it when the square of half r is equal to, or greater than the cube of one-third of q; and that it is of no use in resolving the other case of the same equation, where the square of half r is less than the cube of one-third of q, because then qis a negative quantity, and, consequently, its 4 27 square root is impossible. The Baron begins by first laying down and investigating CARDAN's rule for the resolution of the cubic equation, where that term is wanting that involves the square of the unknown quantity; gives an example of the method of applying this rule to the common cale, in the form of an analytical investigation ; and afterwards, adds a synthetical demonstration of the same thing. |