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thing. He next gives three other expressions for the root of this equation, each of which he both investigates analytically, and demonstrates synthetically, 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 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 Isaac Newton's celebrated binomial theorem, he arrives at laft, 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 Maleres'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 impoflible quantities; and he has taken care to point out, as he went along, under what cire. cumstances the feries which he has occasion to consider 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 those which Dr. Wallis and Mr. Demoivre have given of the same problem : and he concludes his paper with a bitter Philippic against the very general and extensive idea which modern algebraists have annexed to the negative sign.
MECHANICS. 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 Wheel 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
ses Effets que nécessaire pour diterminer les parfaits Rapports entre les differens Mobiles agisans par Leviers et par Engrenages.
A direct and certain method of finding the true diameter of 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 desideratum in the arts of clock and watch ma. king. At first fight it appears, and indeed Mr. Derham * directs that their circumferences, and consequently 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 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 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 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 meafure, 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 theoretically 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 'ic 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 three full teeth of the wheel it is to work with ; for the diame. ter 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 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 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 usefulness of such an instrument will be readily admitted by every watchmaker ; but whether the in
ftrument that Mr. Le Cerf has invented will answer this puro; pose or not, rests solely on the bare assertion of the inventor. He does not pretend that even the proportion, on which its construction depends, is the result of a mathematical investigation, 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 such 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 must afterwards be reduced : and the quantity of that reduction he finds by a rule which is in substance 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 arising 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, expressed in such measures as the diameter of the pinion was taken in. Or this rule may be expressed 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, so is the diameter of the pinion to the reduction of the diameter of the wheel. Thus if the diameter of the pinion be t'ó, its leaves 12, and the teeth of the wheel 96; then the diameter of the, wheel, according to Mr. Der ham, will be to X.96 +12= id; and, according to Mr. Le Cerf, 12 is to 1 -1(=7) as i's: 7 Xit + 12 =110: consequently 0-10=1996 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.
That this mode of reduction may be sufficiently near the truth for mechanical purposes, may possibly be a fact : but that it is strictly true, notwithstanding Mr. Le Cerf's affertions, may admit of a doubt. For aught that appears to the contrary, several other laws of reduction may be assigned that will answer equally well,-- perhaps better. On the whole, it seems that the rule having occurred to Mr. Le Cerf, its simplicity and uniformity pleased him, and the work, formed by ít, happening to work freely, convinced him that it was true : but this is no abfolute proof that it is fą. Mr. Le Cerf's measures may have been taken rather inaccurately; or, possibly, some of his pic nions might have worked ftill better, on a farther, or a less reduction. We do not mean, by what we have said, to depreeiate Mr. Le Cerf's invention; his pursuits are truly laudable and useful, the thought ingenious, and may be true ;-we only with to convince him that he is positive without proof.
The form and construction of Mr. Le Cerf's compasses cannot be gathered with certainty from the paper before us, as his
description of them is very concise, although his paper, in other respects, is diffusive, 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 addrefled, when the instrument 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 translation-we give it up entirely-those may read it who can, But is it not a moft extraordinary circumftance that so 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) Tould suffer such translations as this, and some 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 likewise. Nothing can be more obvious than that when the sides 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 in, veftigate; and, perhaps, not very difficult to work, sufficiently 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 practised by the beit 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 cooth acts on its fellow for a less time; that is, while the wheels move through a less angular space, and of courfe does not act under such a variety of angles as they must 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-spring, which is undoubtedly sometimes much greater, has but little effect on the going of a watch, the balance of which has sufficient momentum, --such as all the watches that are made by the best English artists now have : the figure of the teeth is therefore but of little consequence. In watches of Mr. Harrison's construction these inequalities cannot poffibly have any effe&t, because that part of the watch which meafures time is moved by a small spring that acts on the contrate wheels and, consequently, whatever irregularities there may be in the forces which act on the other wheels, they can noway affect the going of the watch.
forces consumed ;
The Author adds an account of another instrument which he has invented, and fent along with the Proportional Compasses to the Royal Society ; but the vague manner in which he speaks of it, joined to the want of a drawing, renders it impossible for us to form a juft idea either of its construction or merits. He concludes his paper with tables of the dimensions of the several pinions generally used in clock and watch work, ac, cording to the principles which he has before laid down.
CHEMISTRY. Article 39. Chemical Experiments and Observations 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 specific gravities of various lead ores, and even of different parts of the same lump of ore. Notwithstanding this circumstance, we are told that the purchasing of lead ore by the measure, is the general, though not the universal custom in Derbyshire. To find whether the sulphur with which lead is generally mineralised in the ore (particularly in the feelgrained and tellelated galenas) could be separated from it in close vefsels, or by distillation, as is the case with respect to some kinds of the pyrites; he distilled 16 ounces of some telle, Jated 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 separated from the ore were—a small quantity of a black substance, that role up into the neck of the receiver ; and which appeared to be pure lead ore, sublimed without being decompounded :-a small portion of a liquid, that had a pungent smell, resembling that of the volatile vitriolic acid, and which had an acid taste; though it did not effervesce with alcalis, nor produce any change in the colour of blue paper :-and lastly, a quantity of air or elaftic fuid ; which, at the beginning of the process, had the smell of inflammable air. In the following experiment, however, he not only separated the fulphur from the ore, but was enabled to afcertain its quantity.
Five ounces of the strongest fuming spirit of nitre, with an equal quantity of water; were poured on ten ounces of lead ore. A violent effervescence ensued; and when the solu, tion was completed, there remained floating upon the surface of the menftruum, a cake of fine yellow fulphur, perfectly re, fembling com.non 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 some of it on a red hot iron, a greyish calx remains, after the sulphur is