is to the sum of the two contrary attractions of the hill as 87522720 is to 88115; that is, as 9933 to i, very near; observing, that this conclusion is founded on the supposition that the density 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 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 bill, 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 denlity 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 on: of the metals? In this point, the Author observes, any confiderable degree of accuracy can only be obtained by a close examination of the internal Aructure of the mountain : and he thinks that the easiest method of doing this would be by boring holes, in several parts of it, to a sufficient depth, in the same manner that is done in searching for coal mines, and then taking a mean of the densities of the several strata which the tool pafles through, as also of the quantities of matter in each ftratum. 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. Hutton 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 common stone is to that of rain water as 21 to 1; which being compounded 1 1 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 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 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 denfity of the earth to that of water, and the relative densities of the planets one to another, determined from physical considerations, to find their densities relative to rain water, which he makes as follows: Water Mars 1 311 Jupiter 51 Saturn 4 i Mr. Hutton concludes his paper with pointing out some para ticulars which may tend to render the experiment more complete and accurate if it hould ever be repeated. Article 41. A Method of finding, by the Help of Sir Isaac New ton's binomial Theorem, á near Value of the very slowly-converging infinite Series *+**+*+*+&c. when x is very nearly . * 4 equal to 1. By Francis Maseres, Esq; F.R.S. Cursilor 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 manifeft, A being equal 1, B=i, C=, D=i, &c. that B will be equal to LĂ, C=}B, D=1C, and so on; and consequently, by substituting these quantities for their equals in the original series, it will become *+ 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 73 n is equal to the mtn feries it- 4x + Cx3+ 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 2 m m+3Dx++ &c. 2n m Bx+ + 2n 3n n 2 n 4 2 immediately preceding. Hence if n be assumed very great in respect 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 mtn m+an it Cx} + &c. will be very nearly equal to the series 1+4. Ax+ Br? +21.Cxit &c. that is, 31 to it". Ax+iBx? +Cx} + &c. by abbreviation. · Or, by re ". , itoring the values of m, A, B, C, &c. to 1+*+*+ *+*+&c. Hence, therefore, it is evident that the 3n 4 n binomial () :) is very nearly equal to the feries it ; *+ *****"+'&c. and multiplying by nh, n X1-* ;=n+*+-+-+-t &c. and subtracting 3 x? *-n=x+ +*+-t &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, x3*-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 ose 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 " - is a negative quantity, and, consequently, its 4 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 case, in the form of an analytical investigation; and afterwards adds a synthetical demonftration of the same thing. He next gives three other expressions for the root of this equation, each of which he both investigates analytically, and demonstrates fynthetically, and adds an example of resolving this equation by each of them. He then proceeds to thew how the preceding rules may be adapted to the solution of that case of this equation, in which the square of half ? is less than the cube of one-third part of q, or to what is generally called the irreducible cafe. In which, by a happy application of Sir Ifaac Newton's celebrated binomial theorem, he arrives at last, after a long train of algebraical reasoning, at an infinite series, which, as he afterwards thews, being multiplied into the cube root of half r, will give the value of x, the root fought. Baron Maferes's principal view in this paper seems to have been to investigate the solution of this case of a cubic equation, without the confideration of impossible quantities; and he has taken care to point out, as he went along, under what circumstances the series which he has occasion to confider will converge, and when they will not: as also carefully to distinguish in which cases the affirmative and negative signs take place ; so that it will be no difficulty to follow him through the whole of this long and laborious process, if any person thinks proper to take the trouble of doing it. Several examples are added ; and also a scholium, in which he compares his own solution with thofe which Dr. Wallis and Mr. Demoivre have given of the fame problem : and he concludes his paper with a bitter Philippic against the very general and extensive idea which modern algebrailts have annexed to the negative sign. ME.CHANIC S. Article 43. Account of the Advantages of a newly invented Ma chine, much varied in its Effects, and very useful for determining the perfect Proportion between different Moveables acting by Levers and IV bed and Pinion. By Mr. Le Cerf, Watchmaker at Geneva ; communicated by Lord Viscount Mahon, F.R.S. In French (the original) as follows : Description d'une Machine de nouvelle Invention, aussi variée dans jes Effets que nécesjaire pour d terminer les parfaits Rapports entre les differens Mobiles agissans par Leviers et par Engrenages. A direct and certain method of finding the true diameter of a pinion which is to be acted on by a wheel of a given diameter, or the diameter of a wheel which is to drive a pinion of a given diameter, the number of teeth in each being also given, has been hitherto a defideraturn in the arts of clock and watch making. At first fight it appears, and indeed Mr. Derham * dis sects that their circumferences, and consequently their diame. ters, should be as the number of teeth ; but Mr. D. made watches only in theory, and would have found out his mistake • Artificial Clockmaker, p. 69. the first time that he had attempted to put it in practice. If the circumferences of the wheel and pinion were to run against, instead of taking in to one another, this proportion would be juft; but as this is not the case, and that the circumference of one is to enter into and lay hold of that of the her, the proportion is not to be made between the extreme circumference of the pinion and the extreme circumference of the wheel, but between the extreme circumference of the wheel and the circumference of the pinion, at a point somewhat within its extreme circumference : and the distance of this point from the extreme circumference depends jointly on the diameters, and the number of teeth and leaves there are in the wheel and pinion : it depends also, we conceive, in some mea. fure, on the substance, and form of the teeth, although Mr. Le Cerf will not admit of it. No wonder, therefore, that watchmakers, instead of endeavouring to investigate theoretic cally a proportion so complicated, should try to find, mechanically, such practical rules as would readily discover the true diameters nearly, and afterwards reduce them to the true ones by trials. Accordingly, Mr. Le Cerf tells us that watchmakers, in general, proportion the diameters of their pinions to those of the wheels nearly, by taking the extent from the point of any one tooth of a wheel to the point of the next tooth to it except one, or, according to some, a little more than this extent, for the diameter of a pinion of fix leaves which will work in that wheel; for the diameter of a pinion of seven, they take tbree full teeth of the wheel it is to work with ; for the dianieter of a pinion of eight, three teeth and the space between the third and fourth; for a pinion of ten leaves, four full teeth of the wheel as it comes out of the engine; and, lastly, for the diameter of a pinion of twelve leaves, rather more than the extent from the point of one tooth to the point of the fifth tooth from it. The wheel and pinion being finilhed to these dimenfions, they try if they work well together; if they do not, and the pinion be too large, they reduce it until they do; but if the pinion be too small, they have nothing to do but make a larger. Mr. Le Cerf informs us, in the paper under confideration, that he has discovered a direct and simple method of determine ing the true diameter which any pinion ought to have, so thac it may work freely with any wheel of a given diameter and number of teeth ; and from thence has constructed a new instrument, which he calls the Proportional Compafles, by means of which the proper diameter of any wheel to that of a pinion, or of any pinion to a given wheel, may be readily determined, and with the utmost accuracy, let the number of teeth in each be what they will. The utefulness of such an inftrument will be readily admitted by every watchmaker; but whether the in Strumet |