naw under confideration justly obferves, a frequent repetition of the fame experiment, and a coincidence of the refults, afford that firm dependance on the conclufions, and fatisfaction to the mind, which can fcarcely ever be had from a fingle trial, however carefully it may be executed.' The refult of the experiment, as well as the manner of conducting it, is related at large by the Doctor himself in the Philofophical Tranfactions 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 inftrument was deflected from its true perpendicular direction, by the attraction of the mountain, by an angle of about 5 fe conds: the fum of the two deviations being 11". 6; and which eftablishes the truth of the Newtonian philofophy on the solid foundation of experiment. It remained ftili to determine, from this most curious experiment, the ratio of the mean denfity of the hill to that of the earth, and from hence, and the known matter of which the hill confifts, 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 Tranfactions for as the business was in its nature entirely new, it laid Mr. Hutton under the neceflity of inventing, and, of courfe, defcribing at length, the feveral modes of computation which he has made ufe of, and alfo of giving a fynopfis of the measures which were taken of the feveral lines and angles, that any perfon, who thinks proper, may fatisfy himself of the truth of the computations here delivered. It appears that two principal bafes were measured, befide other fhorter lines, one on the fouth, and the other on the north-west fides of the mountain. From these two bafes, and the feveral angles which were alfo measured, both vertical and horizontal, from their feveral extremities to different parts of the fummit and base of the mountain, as well as different points on its furface, the plan of it, as well as the figure of a prodigious multitude of its fections were computed; and from thence alío the figure of the hill was conftructed, on a very large fcale, upon paper. Notwithstanding this ftupendous piece of computation was thus effected, one, not lefs arduous, appeared behind, which was to apply the foregoing calculations and conftructions to the determination of the effect of the attraction of the mountain in the direction of the meridian: and here it foon occurred to the ingenious Computer, that the heft method would be to divide the plan into a great number of fmall parts, which might be conidered as the bafes of fo many fmall columns, or pillars of matter into which the hill and the adjacent ground was divided by vertical planes, forming an imaginary groupe of vertical columns, fomething like a fet of bafaltine pillars, or like the cella in a piece of honey-comb; then to compute the attraction of each pillar feparately in the direction of the meridian; and, laftly, to take the fum of all thefe computed effects for the whole attraction of the matter in the hill. It is obvious that the attraction of any one of thefe pillars, on a body in a given place, may be eafily computed, and that in any direction, becaufe of the smallness and given pofition of its bafe: for on account of its small nefs all the matter in the pillar may be fupposed to be collected into its axis or vertical line, erected on the middle of its bafe, the length of which axis, as the mean altitude of the pillar is to be eftimated 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 pofition of the bafe, and the matter fuppofed to be contained in the pillar, and collected into the axis, a theorem is eafily 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 fmall to admit of this mode of computation, not less than a thousand for each obfervatory, or two thoufand in the whole, forming the bafes of as many fuch pillars of matter as have been defcribed above; which, if the attractions of every one had been feparately computed, muft evidently have been a work of fuch labour as would have difcouraged every perfon from undertaking it; but which muft neverthelefs have been the cafe if our Author had not luckily hit upon a method of dividing his matter into columns, fo 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 procefs by means of which it is derived, cannot poffibly be pointed out here: fuffice it to fay, that the refult of this long and intricate calculation was, that the effect of the attraction of the matter in the mountain and adjacent hills, at the fouthern obfervatory, was to the effect of the fame attraction at the northern one, as 69967 to 88644, or as 7 to 9 very near. This difference, Mr. Hutton fhews, is to be attributed principally to the effect of the hills which lie on the fouth fide of the mountain Schehallien, and which are not only larger, but also nearer to it than those which are on its north fide. 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 fum of the two contrary attractions of the hill as 87522720 is to 8811; that is, as 9933 to I, very near; obferving, that this conclufion is founded on the fuppofition that the denfity of the matter in the hill is equal to the mean denfity of all the matter in the earth. But the Aftronomer Royal found, by his obfervations, that the fum of the deviations of the plumb-line produced by the two contrary attractions of the mountain was 11". 6: from which circumftance it may be inferred, that the attraction of the earth is actually to the fum of the two contrary attractions of the hill, as radius to the tangent of 11". 6, nearly; that is, as 1 to .000056239, or as 17781 to 1. Or, after allowing for the centrifugal force arifing from the rotation of the earth about its axis, as 17804 to nearly. Having thus obtained the ratio which actually exifts between the attraction of the whole earth and that of the mountain, refulting from the obfervations, and alfo the ratio of the fame things arifing from the computation, on the fuppofition of an equal denfity; 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 feveral circumftances 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 queftion yet remains to be determined, namely, what is the proportion between the density of the matter in the hill, and that of fome known fubftance; for example, water, ftone, or fome one of the metals? In this point, the Author obferves, any confiderable degree of accuracy can only be obtained by a clofe examination of the internal ftructure of the mountain: and he thinks that the eafieft me→ thod of doing this would be by boring holes, in feveral parts of it, to a fufficient depth, in the fame manner that is done in fearching for coal-mines, and then taking a mean of the denfities of the several ftrata which the tool paffes through, as alfo of the quantities of matter in each ftratum, But as this has not been done, we must reft fatisfied with the estimate arifing from the report of the external view of the hill, which, to all appearance, confifts of an entire mafs of folid rock: Mr. Hut ton thinks, therefore, that he will not greatly err by affuming the denfity of the hill equal to that of common ftone, which is not much different from the mean denfity of the whole matter, near the earth's furface, to fuch depths as have hitherto been explored, either by digging or boring. Now the density of common ftone is to that of rain-water as 2 to 1; which being compounded compounded with the proportion of 9 to 5, found above, gives 4 to 1 for the ratio of the mean denfity of the whole earth to that of rain-water. Sir Ifaac Newton thought it probable that this proportion might be about 5 or 6 to 1: fo much justness was there even in the furmises of this wonderful man! Mr. Hutton proceeds to obferve, that as the mean denfity of the earth is about twice the denfity of the matter near the furface, there must be somewhere, towards the more central parts, great quantities of metals, or other very denfe fubftances, to counterbalance the lighter materials, nearer to the furface, and produce fuch a confiderable mean denfity. He then goes on, having the ratio of the mean denfity of the earth to that of water, and the relative denfities of the planets one to another, determined from phyfical confiderations, to find their denfities relative to rain water, which he makes as follows: Water The Sun Mercury I Mars 4 / 3루 3TT Venus The Earth 1 3 32 Mr. Hutton concludes his paper with pointing out fome 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 Ifaac Newton's binomial Theorem, a near Value of the very flowly-converging XX X3 X4 infinite Series x++*+*+&c. when x is very nearly 2 3 4 equal to 1. By Francis Maferes, Efq; 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 feries, it is manifeft, A being equal 1, B, C=y, D=}, &c. that B will be equal to A, C=}B, D={C, and fo on; and confequently, by fubftituting thefe quantities for their equals in the original feries, it will become x+x+}Bx3+¿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 alfo be found, by Sir Ifaac Newton's theorem, that the binomial 1-x) m+2n 772 " is equal to the 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 conftituted by adding n both to the numerator and denominator of the co-efficient of the term immediately immediately preceding. Hence if n be affumed very great in refpect of m: for example if m be taken =1, and none billion, or the fquare of one million, it is evident that the feries Ax Bx2+m+2n Cx2+ &c. will be very nearly m n + m+n 2 n 3n 2n n_Bx2+ equal to the series 1+Ax+. -Bx2 + 2 Cx3+ &c. that is, m n 2n 3n to 1+ Ax+Bx2+Cx1+ &c. by abbreviation. Or, by re n ftoring the values of m, A, B, C, &c. to 1+-x+, 2n I x++ &c. Hence, therefore, it is evident that the 4 n I 3 n m the series 1+ - *+ —2++ &c. and multiplying by n, n × 1—3——=n+x+*+*+*+ &c. and fubtracting 1−xì—=—n=x+*+*+*+ &c. n being 1 with 2 3 4 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 refolving one Cafe of a cubic Equation of this Form, x3-qx=r, to the other Cafe of the fame Equation, which it is not naturally fitted to folve, and which is therefore often called the irreducible Cafe. By Francis Maferes, Efq; F. R. S. Curfitor Baron of the Exchequer. It is well known to all perfons converfant with algebra, that Cardan's rule for folving the cubic equation, x3-qx=r, is only fuited to refolve it when the fquare of half r is equal to, or greater than the cube of one-third of q; and that it is of no ufe in refolving the other cafe of the fame equation, where the fquare of half r is less than the cube of one-third of 9, because then is a negative quantity, and, confequently, its The Baron begins by firft laying down and inveftigating CARDAN'S rule for the refolution of the cubic equation, where that term is wanting that involves the fquare of the unknown quantity; gives an example of the method of applying this rule to the common cafe, in the form of an analytical inveftigation; and afterwards adds a fynthetical demonftration of the fame |