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 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 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 I 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 some known fubstance; for example, water, ftone, or fome on of the metals? In this point, the Author obferves, any confiderable degree of accuracy can only be obtained by a close examination of the internal ftructure of the mountain: and he thinks that the eafieft method 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 searching for coal mines, and then taking a mean of the denfities of the feveral ftrata which the tool paffes through, as alfo of the quantities of matter in each ftratum. Bet as this has not been done, we must reft fatisfied with the eftimate arifing, from the report of the external view of the hill, which, to all appearance, confifts of an entire mafs of folid rock: Mr. Hutton 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 denfity 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 juftness was there even in the furmifes of this wonderful man!

Mr. Hutton proceeds to observe, that as the mean denfity of the earth is about twice the denfity of the matter near the furface, there must be fomewhere, towards the more central parts, great quantities of metals, or other very dense substances, 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:

3루

31T

124

13

32

The Earth

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 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 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 and its powers in the above feries, it is manifeft, A being equal 1, B, C=}, D=}, &c. that B will be equal to 4, C=3B, D=IC, and fo on; and confequently, by fubftituting thefe quantities for their equals in the original feries, it will become x+4x2+3Bx1+3Ċx++&c. where it may be obferved that the fractional, or numerical part of the co-efficient of each term, after the firft, is derived by adding I both to the numerator and denominator. It will alfo be found, by Sir Ifaac

n

m+n

2 ከ

Bx2+m+2n
3 n

m+3′′ Dx++ &c.

12

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 m+n Bx2+m+2n Cx2+ &c. will be very nearly

m

Ax

n

3 n

2 Cx &c, that is,

2 n

equal to the series 1+Ax+

m

n

3"

to + Ax+ 1⁄2 Bx2+3Cx3+ &c. by abbreviation. Or, by re

I

n

ftoring the values of m, A, B, C, &c. to 1+-x+,

2n

+

3 n

binomial

4 n

x++ &c. Hence, therefore, it is evident that the

(-1)) is very nearly equal to

the feries 1+++*2+++3+'&c. and multiplying by

», » × 1−x¬==n+x+*+*+*+ &c. and subtracting

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 q, because then is a negative quantity, and, confequently, its

Tr

4 27 fquare root is impoffible.

The Baron begins by firft laying down and investigating 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 investigation; and afterwards adds a fynthetical demonftration of the fame

thing.

thing. He next gives three other expreffions for the root of this equation, each of which he both inveftigates analytically, and demonftrates fynthetically, and adds an example of refolving this equation by each of them. He then proceeds to fhew how the preceding rules may be adapted to the solution of that cafe of this equation, in which the fquare of half r 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 laft, after a long train of algebraical reasoning, at an infinite feries, which, as he afterwards fhews, 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 feems to have been to investigate the folution of this cafe of a cubic equation, without the confideration of impoffible quantities; and he has taken care to point out, as he went along, under what circumstances the feries which he has occafion to confider will converge, and when they will not: as alfo carefully to diftinguish in which cafes the affirmative and negative figns take place; fo that it will be no difficulty to follow him through the whole of this long and laborious procefs, if any perfon thinks proper to take the trouble of doing it. Several examples are added; and alfo a fcholium, in which he compares his own folution 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 extenfive idea which modern algebraifts have annexed to the negative fign.

MECHANICS.

Article 43. Account of the Advantages of a newly invented Machine, much varied in its Effects, and very useful for determining the perfect Proportion between different Moveables acting by Levers and Wheel and Pinion. By Mr. Le Cerf, Watchmaker at Geneva; communicated by Lord Viscount Mahon, F. R. S. In French (the original) as follows:

Defcription d'une Machine de nouvelle Invention, auffi variée dans fes Effets que néceffaire pour determiner les parfaits Rapports entre les differens Mobiles agiffans 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 alfo given, has been hitherto a defideratum in the arts of clock and watch making. At first fight it appears, and indeed Mr. Derham* directs that their circumferences, and confequently their diameters, fhould 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

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 cafe, and that the circumference of one is to enter into and lay hold of that of the other, 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 fomewhat 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 alfo, we conceive, in fome meafure, on the fubftance, and form of the teeth, although Mr. Le Cerf will not admit of it. No wonder, therefore, that watchmakers, instead of endeavouring to inveftigate theoretically a proportion fo complicated, fhould try to find, mechanically, fuch practical rules as would readily difcover 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 thofe 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 fome, 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 feven, they take three full teeth of the wheel it is to work with; for the diameter 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 finished to thefe 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 fmall, they have nothing to do but make a larger. Mr. Le Cerf informs us, in the paper under confideration, that he has difcovered a direct and fimple method of determin ing the true diameter which any pinion ought to have, fo that it may work freely with any wheel of a given diameter and number of teeth; and from thence has conftructed a new inftrument, which he calls the Proportional Compaffes, 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 ulefulness of fuch an inftrument will be readily admitted by every watchmaker; but whether the in

ftrument