thing. He next gives three other expreffions for the root of this equation, each of which he both investigates 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 folution 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 reafoning, 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 circumftances 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 PhiHippic 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, 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

the first time that he had attempted to put it in practice. If the circumferences of the wheel and pinion were to run againft, instead of taking in to one another, this proportion would be just; 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 diftance 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 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 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, laftly, 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 finifhed 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 simple method of determining the true diameter which any pinion ought to have, so 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 ufefulness of fuch an inftrument will be readily admitted by every watchmaker; but whether the in

ftrument

ftrument that Mr. Le Cerf has invented will answer this purpofe or not, refts folely on the bare affertion of the inventor. He does not pretend that even the proportion, on which its conftruction depends, is the refult of a mathematical inveftigation, but that it is only derived from experiments; by means of which he has found, that if the diameter of any wheel be made in fuch proportion to that of the pinion it is to work with, as the number of teeth in the wheel is to the number of leaves in the pinion, that wheel will be too large; and its diameter muft afterwards be reduced: and the quantity of that reduction he finds by a rule which is in fubftance as follows. Subtract unity from the number of revolutions which the pinion makes in one revolution of the wheel, and multiply the remainder by the quotient arifing from dividing the diameter of the pinion by the number of leaves which are in it: the product will express the quantity by which the diameter of the wheel is to be leffened, expreffed in fuch measures as the diameter of the pinion was taken in. Or this rule may be expreffed by the following analogy: As the number of leaves in the pinion is to the excess of the number of revolutions which the pinion makes, above that which is made by the wheel, fo is the diameter of the pinion to the reduction of the diameter of the wheel. Thus if the diameter of the pinion be to, its leaves 12, and the teeth of the wheel 96; then the diameter of the wheel, according to Mr. Derham, will be to X.96 ÷ 12 = 10; and, according to Mr. Le Cerf, 12 is to 2-1 (= 7) as 15: 7X16÷12=12: confequently 18-120-12% will be the true diameter of a wheel of 96 teeth, which is proper to work with a pinion of 12 leaves, and one-tenth of an inch diameter.

7

8

That this mode of reduction may be fufficiently near the truth for mechanical purposes, may poffibly be a fact: but that it is ftrictly true, notwithstanding Mr. Le Cerf's affertions, may admit of a doubt. For aught that appears to the contrary, feveral other laws of reduction may be affigned that will answer equally well,-perhaps better. On the whole, it seems that the rule having occurred to Mr. Le Cerf, its fimplicity and uniformity pleafed him, and the work, formed by it, happening to work freely, convinced him that it was true: but this is no abfolute proof that it is fo. Mr. Le Cerf's measures may have been taken rather inaccurately; or, poffibly, fome of his pinions might have worked ftill better, on a farther, or a lefs reduction. We do not mean, by what we have faid, to depreeiate Mr. Le Cerf's invention: his purfuits are truly laudable and ufeful, the thought ingenious, and may be true; we only wish to convince him that he is pofitive without proof.

The form and conftruction of Mr. Le Cerf's compaffes cannot be gathered with certainty from the paper before us, as his defcription

2.

defcription of them is very concife, although his paper, in other refpects, is diffufive; and, which is a much greater defect, there is no drawing of them annexed to it: therefore, although his matter might appear very clear to the learned Society to whom it is addreffed, when the inftrument was before them, yet we apprehend that few workmen will be able to gather much information, even from the original French. As to the English tranflation-we give it up entirely-thofe may read it who can. But is it not a moft extraordinary circumftance that fo learned a body of men as the Council of the Royal Society may be supposed to confift of (for the Society at large we well know have no concern in it) fhould fuffer fuch tranflations as this, and fome other late ones, to appear in their publication?

Mr. Le Cerf, in the course of his paper, takes occafion to mention the form which the teeth ought to have, so that one wheel, moving uniformly, may drive another with an uniform velocity likewife. Nothing can be more obvious than that when the fides of the teeth are planes, or nearly so, if the driving wheel has an uniform motion round its center, the motion of the wheel which is driven by it will be very unequal; moving with great velocity when any tooth first begins to act on it, and scarcely at all when the planes of the two teeth make a great angle with each other. The figures which the faces of the teeth of the driven wheel ought to have, in order that both wheels may move uniformly, is not difficult to inveftigate; and, perhaps, not very difficult to work, fufficiently, near the truth, were it an object of importance enough to merit it. This, however, Mr. Le Cerf has not attempted to do, but advises a method that has long been practifed by the best English watchmakers, namely, putting the highest numbers in the wheels and pinions that the caliber of the watch will admit of. For it is evident, that by this means, any fingle tooth acts on its fellow for a lefs time; that is, while the wheels move through a lefs angular space, and of courfe does not act under fuch a variety of angles as they muft unavoidably do when the numbers in the wheels and pinions are lower. We may add, that the unequal action of the teeth, or even that of the main-fpring, which is undoubtedly fometimes much greater, has but little effect on the going of a watch, the balance of which has fufficient momentum,-fuch as all the watches that are made by the best English artists now have: the figure of the teeth is therefore but of little confequence. In watches of Mr. Harrifon's conftruction thefe inequalities cannot poffibly have any effect, becaufe that part of the watch which meatures time. is moved by a small fpring that acts on the contrate wheel, and, confequently, whatever irregularities there may be in the

forces

forces which act on the other wheels, they can noway affect the going of the watch.

The Author adds an account of another inftrument which he has invented, and fent along with the Proportional Compaffes to the Royal Society; but the vague manner in which he speaks of it, joined to the want of a drawing, renders it impoffible for us to form a juft idea either of its conftruction or merits. He concludes his paper with tables of the dimenfions of the feveral pinions generally ufed in clock and watch work, according to the principles which he has before laid down.

CHEMISTRY.

Article 39. Chemical Experiments and Obfervations on Lead Ore. By Richard Watson, D. D. F. R. S. &c.

In this paper, Dr. Watson first takes notice of the difference in the fpecific gravities of various lead ores, and even of different parts of the fame lump of ore. Notwithstanding this circumftance, we are told that the purchafing of lead ore by the measure, is the general, though not the univerfal custom in Derbyshire. To find whether the fulphur with which lead is generally mineralifed in the ore (particularly in the fteelgrained and teffelated galenas) could be feparated from it in clofe veffels, or by diftillation, as is the cafe with respect to fome kinds of the pyrites; he diftilled 16 ounces of fome teflelated Derbyshire lead ore in an earthen retort. Though he gave the retort a white heat, no fulphur was fublimed: but the ore loft a 32d part of its weight. The matters feparated from the ore were-a finall quantity of a black fubftance, that rose up into the neck of the receiver; and which appeared to be pure lead ore, fublimed without being decompounded:-a fmall portion of a liquid, that had a pungent fmell, refembling that of the volatile vitriolic acid, and which had an acid tafte though it did not effervefce with alcalis, nor produce any change in the colour of blue paper :-and laftly, a quantity of air or elaftic fluid; which, at the beginning of the process, had the smell of inflammable air. In the following experiment, however, he not only feparated the fulphur from the ore, but was enabled to afcertain its quantity.

e;

Five ounces of the strongest fuming fpirit of nitre, diluted with an equal quantity of water, were poured on ten ounces of lead ore. A violent effervefcence enfued; and when the folu tion was completed, there remained floating upon the surface of the menftruum, a cake of fine yellow fulphur, perfectly refembling common fulphur. This fubftance, edulcorated and dried, generally amounted to one-third of the weight of the ore,

This matter however, is not pure fulphur, but is a mixture of that substance and a calx of lead: for on putting fome of it on a red hot iron, a greyish calx remains, after the fulphur is confumed;