, or Adfected, Equation, in Algebra, is an equation in which the unknown quantity rises to two or more several powers or degrees. Such, for example, is the equation , in which there are three different powers of x, namely, x3, x2, and x.

The term, affected, is also used sometimes in algebra, when speaking of quantities that have co-efficients.

Thus in the quantity 2a, a is said to be affected with the co-efficient 2.

It is also said, that an algebraic quantity is affected with the sign + or -, or with a radical sign; meaning no more than that it has the sign + or -, or that it includes a radical sign.

The term adfected, or affected, I think, was introduced by Vieta.

, in Geometry, a term used by some ancient writers, signifying the same as property.

Affection. Phys. The affections of a body are certain modifications occasioned or induced by motion; in virtue of which the body is disposed after such, or such a manner.

The affections of bodies, are sometimes divided into primary and secondary.

Primary Affections, are those which arise either out of the idea of matter, as magnitude, quantity, and figure; or out of the idea of form, as quality and power; or out of both, as motion, place, and time.

Secondary, or derivative Affections, are such as arise out of primary ones, as divisibility, continuity, contiguity, &c, which arise out of quantity; regularity, irregularity, &c, which arise out of figure, &c.

AFFIRMATIVE Quantity, or Positive Quantity, one which is to be added, or taken effectively; in contradistinction to one that is to be subtracted, or taken defectively.—The term affirmative was introduced by Vieta.

Affirmative Sign, or Positive Sign, in Algebra, the sign of addition, thus marked +, and is called plus, or more, or added to. When set before any single quantity, it serves to denote that it is an affirmative or a positive quantity; when set between two or more quantities, it denotes their sum, shewing that the latter are to be added to the former. So + 6, and + a, and + AB, are affirmative quantities; also + 6 + 8 + 10 denote the sum of 6, 8, and 10, which is 24, and are read thus, 6 plus 8 plus 10. Also a + b + c denote the sum of the quantities represented by a, b and c, when added together. It seems now not easy to ascertain with certainty, when, or by whom, this sign was first introduced; but it was probably by the Germans, as I find it first used by Stifelius in his Arithmetic, printed in 1544.

The early writers on Algebra used the word plus in Latin, or piu in Italian, for addition, and afterwards the initial p only, as a contraction; like as they used minus, or meno, or the initial m only, for subtraction: and thus these operations were denoted in Italy by Lucas de Burgo, Tartalea, and Cardan, while the signs + and - were employed much about the same time in Germany by Stifelius, Scheubelius, and others, for the same operations.

, in Chronology, is used for a century, being a system or a period of a hundred years.

Chronologists also divide the time since the creation of the world into three ages: The first, from Adam till Moses, which they call the age of nature; the second from Moses to Jesus Christ, called the age of the law; and the third, or age of grace, from Jesus Christ till the end of the world.

Age of the Moon, in Astronomy, is the number of days elapsed since the last new moon. To find the moon's age, for any time nearly, for ordinary uses; add together the epact, the day of the current month, and the number of months from March to the present month inclusive; the sum is the moon's age: but if the sum exceed 29, deduct 29 from it in months that have 30 days, or 30 in those that have 31; and the remainder will be the age.—At the end of 19 years the moon's age returns upon the same day of the month, but falls a little short of the same hour of the day.

, Agens, in Physics, that by which a thing is done or effected; or any thing having a power by which it acts on another, called the patient, or by its action induces some change in it.

, the sum or result of several things added together. See Sum.

, in Physics, a brisk intestine motion, excited among the particles of a body. Thus fire agitates the subtlest particles of bodies. Fermentations, and effervescences, are produced by a brisk agitation of the particles of the fermenting body.

, or Aquilon, was a jesuit of Brussels, and professor of philosophy at Doway, and of theology at Antwerp. He was one of the first that introduced mathematical studies into Flanders. He wrote a large work on Optics, in 6 books, which was published in folio, at Antwerp, in 1613; and a treatise of Projections of the Sphere. He promised also to treat upon Catoptrics and Dioptrics, but this was prevented by his death, which happened at Seville, in the year 1617.

, in Physics, a thin, fluid, transparent, elastic, compressible, and dilatable body, which surrounds this terraqueous globe, and covers it to a considerable height.

Some of the ancients considered air as an element, namely, one of the four elements, air, earth, water, and fire, of which they conceived all bodies to be composed; and though it be certain that air, taken in the common acceptation, be far from the simplicity of an elementary substance, yet some of its parts may properly be so called. So that air may be distinguished into proper or elementary, and vulgar or heterogeneous.

Elementary Air, or Air properly so called, is a subtle, homogeneous, elastic fluid; being the basis, or fundamental ingredient, of the whole air of the atmosphere, from which it takes its name. And in this sense Dr. Hales, and other modern philosophers, consider it as entering into the composition of most, or perhaps all bodies; existing in them in a solid state, devoid of its elasticity, and most of its distinguishing properties, and serving as their cement; but, by certain processes, capable of being disengaged from them, recovering its elasticity, and resembling the air of our atmosphere.

The particular nature of this aerial matter we know but little about: what authors have said concerning it being chiefly conjectural. There is no way of examining air pure and defecated from the several matters with which it is mixed; and consequently we cannot pronounce what are its peculiar properties, abstractedly from other bodies.

Dr. Hook, and some others, maintain that it is the same with the ether, or that imaginary fine, fluid, active matter, conceived to be diffused through the whole expanse of the celestial regions: which comes to much the same thing as Newton's subtle medium, or spirit. In this sense it is supposed to be a body sui generis, incorruptible, immutable, incapable of being generated, but present in all places, and in all bodies.

Other philosophers place its essence in elasticity, making that its distinctive character. These suppose that it may be generated, and that it is nothing else but the matter of other bodies, rendered by the changes it has undergone, susceptible of a permanent elasticity. Mr. Boyle produces a number of experiments, which he made on the production of air, that is, according to him, the extraction of a sensible quantity of air from a body in which there appeared to be little or none at all, by whatever means this may be effected. He observes that among the different methods for this purpose, the chief are fermentation, corrosion, dissolution, decomposition, ebullition of water and other fluids, the reciprocal action of bodies, especially saline ones, upon one another; he adds, that different solid and mineral bodies, in the parts of which no elasticity could be conceived to exist, being plunged into corrosive mediums, which also are quite unelastic, will, by the attenuation of their parts from their mutual collision, produce a considerable quantity of elastic air.

Sir Isaac Newton is of the same opinion, according to whom the particles of a dense compact fixed substance, adhering to each other by a powerful attractive force, cannot be separated but by a violent heat, and perhaps never without fermentation; and these bodies, raresied by heat and fermentation, are finally transformed into a truly permanent elastic air. On these principles, he adds, gunpowder produces air on explosion. Optics, Qu. 31, &c.

Common, or heterogeneous Air, is an assemblage of corpuscles of various kinds, which together constitute one fluid mass, in which we live and move, and which we constantly breathe; which compound mass altogether, is called the atmosphere.

In this popular and extensive meaning of the term, Mr. Boyle acknowledges that air is the most heterogeneous body in the universe; and Boerhaave proves that it is an universal chaos, a mere jumble of all species of created things. Besides the matter of light or fire, which continually flows into it from the celestial bodies, and perhaps the magnetic effluvia of the earth, whatever fire can volatilize must be found in the air.

Hence, for instance, 1. All sorts of vegetable matter must be contained in the air; being either exhaled from plants growing all over the face of the earth, or rendered volatile by putrefaction, not excepting even the more solid and vascular parts of them.

2. It is no less certain that the air must contain particles of every substance belonging to the animal kingdom. For the copious emanations which are perpetually issuing from the bodies of animals, in the perspiration constantly kept up by the vital heat, are absorbed by the air; and in such quantities too, during the course of an animal life, that, could they be recollected, they would be sufficient to compose a good round number of the like animals. And besides, when a dead animal continues exposed to the air, all its particles evaporate, and are quickly dissipated; so that the substance which composed the animal, is almost wholly incorporated with the air.

3. The whole fossil kingdom must necessarily be found in the atmosphere; for all of that kind, as salts, sulphurs, stones, metals, &c, are convertible into fume, and must consequently take place among aerial substances. Gold itself, the most fixed of all natural bodies; is found among ores, closely adhering to sulphurs in mines, and so is raised along with the mineral.

Of all the emanations which float in the vast ocean of the atmosphere, perhaps the principal are such as consist of saline particles. Many writers suppose that they are of a nitrous kind; but it is probable that they are of all sorts, as vitriol, alum, mariue salt, and many others. And Mr. Boyle thinks that there may be great quantities of compound salts, not to be met with on or in the bowels of the earth, formed by the fortuitous concourse and mixture of different saline spirits.

We often find the window-glass of old buildings corroded, as if eaten by worms; though we know of no particular salt that is capable of producing such an effect.

Sulphurs too must make a considerable portion of this compound mass, on account of the many volcanos, grotts, caverns, and mines, dispersed over the face of the globe.

Finally, the various attritions, separations, dissolutions, and other mutual operations of matter of different sorts upon one another, may be regarded as the sources of many other neutral, or anony mous bodies, unknown to us, which rise and float in the air.

Air, taken in this extensive sense, is one of the most general and considerable agents in nature; being concerned in the preservation of animal and vegetable life, and in the production of most of the phenomena that take place in the material world.

Its properties and effects, having been the principal objects of the researches and discoveries of modern philosophers, have been reduced to precise laws and demonstrations, forming no inconsiderable branch of mixed mathematics, under the titles of Pneumatics, Aerometry, &c.

Mechanical Properties and Effects of Air. Of these the most considerable are its fluidity, its weight, and its elasticity.

1. Its Fluidity.—The great fluidity of the air is manifest from the great facility with which bodies traverse it; as in the propagation of, and easy conveyance it affords to, sounds, odours and other effluvia and emanations that escape from bodies: for these effects prove that it is a body whose parts give way to any force, and in yielding are easily moved amongst themselves; which is the definition of a fluid. That the air is a fluid is also proved from this circumstance, that it is found to exert an equal pressure in all directions; an effect which could not take place otherwise than from its extreme fluidity. Neither has it been found that the air can be deprived of this property, whether it be kept for many years together consined in glass vessels, or be exposed to the greatest natural or artificial cold, or condensed by the most powerful pressure; for in none of these circumstances has it ever been reduced to a solid state. It is true indeed that real permanent air may be extracted from solid bodies, and may also be absorbed by them; and we also know that in this case it must be exceedingly condensed, and reduced to a bulk many hundred times less than in its natural state: but in what form it exists in those bodies, or how their particles are combined together, is a mystery which remains hitherto inexplicable.

Those philosophers who, with the Cartesians, make fluidity to consist in a perpetual intestine motion of the parts, think they can prove that this character belongs to air: thus, in a darkened room, where the represen- tations of external objects are introduced by a single ray, the corpuscles with which the air is replete, are seen to be in a continual fluctuation. Some moderns attribute the fluidity of the air, to the fire which is intermixed with it; without which, say they, the whole atmosphere would harden into a solid impenetrable mass: and indeed it must be allowed that the more fire it contains, the greater will its fluidity, mobility, and permeability be; and according as the different positions of the sun augment or diminish the degree of fire, the air always receives a proportional temperature, and is kept in a continual reciprocation.

2. Its Weight or Gravity.—The weight or gravity of the air, is a property belonging to it as a body; for gravity is a property essential to matter, or at least a property found in all bodies. But independent of this, we have many direct proofs of its gravity from sense and experiment: thus, the hand laid close upon the end of a vessel, out of which the air is drawn at the other end, soon feels the load of the incumbent atmosphere: thus also, thin glass vessels, exhausted of their air, are easily crushed to pieces by the weight of the external air: and so two hollow segments of a sphere, 4 inches in diameter, exactly fitting each other, being emptied of air, are, by the weight of the ambient air, pressed together with a force which requires the weight of 188 pounds to separate them; and that they are thus forcibly held together by the pressure of the air, is made evident by suspending them in an exhausted receiver, for then they quickly separate of themselves, and fall asunder. Again, if a tube, close at one end, be filled with quicksilver, and the open end be immerged in a bason of the same fluid, and so held upright, the quicksilver in the tube will be kept raised up in it to the height of about 30 inches above the surface of that in the bason, being supported and balanced by the pressure of the external air upon that surface: and that this is the cause of the suspension of the quicksilver in the tube, is made evident by placing the whole apparatus under the receiver of an air-pump; for then the fluid will descend in the tube in proportion as the receiver is exhausted of its air; and then on gradually letting in the air again, the quicksilver reascends to its former beight in the tube: and this is what is called, from its inventor, the Terricellian experiment. Nay farther, air can actually be weighed like any other body: for a rigid vessel, full even of common air, by a nice balance is found to weigh more than when the air is exhausted from it; and the essect is proportionally more sensible, if the vessel be weighed full of condensed air, and more still if it be weighed in a receiver void of air.

But although we have innumerable proofs of the gravitating property of the air, yet the full discovery of the laws and circumstances of it are certainly due to the moderns. It cannot indeed be denied, that several of the ancients had some confused notions about this property: thus Aristotle says that all the elements have gravity, and even air itself; and as a proof of it, says that a bladder inflated with air, weighs more than the same when empty; and Plutarch and Stobæus quote him as teaching that the air in its weight is between that of fire and of earth; and farther, he himself, treating of respiration, reports it as the opinion of Empedocles, that he ascribes the cause of it to the weight of the air, which by its pressure forces itself into the lungs; and much in the same way are the sentiments of Asclepiades expressed by Plutarch, who represents him as saying, among other things, that the external air, by its weight, forcibly opened its way into the breast. But nevertheless it is certain, however unreasonable it may seem, that Aristotle's followers departed in this instance from their master, by asserting the contrary for many ages together. Indeed many of the phenomena arising from this property, have been remarked from the highest antiquity. Many centuries since, it was known that by sucking the air from an open pipe, having its extremity immersed in water, this fluid rises above its level, and occupies the place of the air. In consequence of such observations, sucking pumps were contrived, and various other hydraulic machines; as Heron's syphons, described in his Spiritalia or Pneumatics, and the watering pots known in Aristotle's time under the name of clepsydræ, which alternately stop or run as the singer closes or opens their upper orifice. Indeed the reason assigned, by philosophers many ages after, for this phenomenon, was a pretended horror that nature conceives for a vacuum, which, rather then endure it, makes a body ascend contrary to the powerful solicitation of its gravity. Even Galileo, with all his sagacity, could not for some time hit upon any thing more satisfactory; for he only assigned a limit to this dread of vacuity: having observed that sucking pumps would not raise water higher than 16 brasses, or 34 English feet, he limited this abhorring force of nature, to one that was equivalent to the weight of a column of water 34 feet high, on the same base as the void space. Consequently he pointed out a way of making a vacuum, by means of a hollow cylinder, whose piston is charged with a weight sufficient to detach it from the close bottom turned upwards: this effort he called the measure of the force of vacuity, and made use of it for explaining the cohesion of the parts of bodies.

Galileo however was well apprised of the weight of the air as a body: in his Dialogues he shews two ways of demonstrating it, by weighing it in bottles: the transition was easy from one discovery to another: yet still Galileo's knowledge of the matter was imperfect, that is, as to the particular instance of the suspension of a fluid above its level, by the pressure of the external air.

At length Torricelli fell upon the lucky guess, that the counterpoise which keeps fluids above their level, when nothing presses upon their internal surface, is the mass of air resting upon the external one. He discovered it in the following manner: In the year 1643 this disciple of Galileo, on occasion of executing an experiment on the vacuum formed in pumps, above the column of water, when it exceeds 34 feet, thought of using some heavier fluid, such as quicksilver. He conceived that whatever might be the cause by which a column of water of 34 feet high is sustained above its level, the same force would sustain a column of any other fluid, which weighed as much as that column of water, on the same base; whence he concluded that quicksilver, being about 14 times as heavy as water, would not be sustained higher than 29 or 30 inches. He therefore took a glass tube of several feet in length, sealed it hermetieally at one end, and filled it with quicksilver; then inverting it, and holding it upright, by pressing his finger against the lower or open orifice, he immersed that end in a vessel of quicksilver; then removing his singer, and suffering the sluid to run out, the event verisied his conjecture; the quicksilver, faithful to the laws of hydrostatics, descended till the column of it was about 30 inches high above the surface of that in the vessel below. And hence Torricelli concluded that it was no other than the weight of the air incumbent on the surface of the external quicksilver, which counterbalanced the fluid contained in the tube.

By this experiment Torricelli not only proved, what Galileo had done before, that the air had weight, but also that it was its weight which kept water and quicksilver raised in pumps and tubes, and that the weight of the whole column of it was equal to that of a like column of quicksilver of 30 inches high, or of water 34 or 35 feet high; but he did not ascertain the weight of any particular quantity of it, as a gallon, or a cubic foot of it, nor its specific gravity to water, which had been done by Galileo, though to be sure with no great accuracy, for he only proved that water was more than 400 times heavier than air.

Torricelli's experiment became famous in a short time. Father Mersenne, who kept up a correspondence with most of the literati in Italy, was informed of it in 1644, and communicated it to those of France, who presently repeated the experiment: Messrs. Pascal and Petit made it first, and varied it several ways; which gave occasion to the ingenious treatise which Pascal published at 23 years of age, intitled Experiences Nouvelles touchant la Vuide. In this treatise indeed he makes use of the old principle of suga vacui; but afterwards getting some notion of the weight of the air, he soon adopted Torricelli's idea, and devised several experiments to consirm it. One of these was to procure a vacuum above the reservoir of quicksilver; in which case he found the column sink down to the common level: but this appearing to him not sufficiently powerful to dissipate the prejudices of the ancient philosophy, he prevailed on M. Perier, his brother-in-law, to execute the famous experiment of Puy-de-Domme, who found that the height of the quicksilver half-way up the mountain was less, by some inches, than at the foot of it, and still less at the top: so that it was now put out of doubt that it was the weight of the atmosphere which counterpoised the quicksilver.

Des Cartes too had a right notion of this effect of the air, to sustain fluids above their level, as appears by some of his letters about this time, and some years before; and in one of those he lays claim to the idea of the Puy-de-Domme experiment: After having desired M. de Carcavi to inform him of the success of that experiment, which public rumour had advertised him had been made by M. Pascal himself, he adds, “I had reason to expect this from him, rather than from you, because I first proposed it to him two years since, assuring him at the same time, that although I had not tried it, yet I could not doubt of the consequence; but as he is a friend of M. Roberval, who professes himself no friend to me, I suppose he is guided by that gentleman's passions.” See more of this history under Barometer.

As to the actual weight of any given portion of common air, it seems that Galileo was the first who determined it experimentally; and he gives two different methods, in his Dialogues, for weighing it in bottles: he did not however perform the experiment very accurately, as he stated from the result that the gravity of water was to that of air rather above 400 to 1.

A quantity of air was next weighed by Mersenne in a very ingenious manner. His idea was to weigh a vessel both when full of air, and when emptied of it a to make the vacuum for this purpose, he knew no better way than by expelling the air out of an colipile by heating it red hot: by weighing it both when cold and hot, he sound a certain difference; which however was not the exact weight of that capacity of air, because the vacuum was not perfect. But by plunging the eolipile, when red hot, into water, just so much water entered as was equal in bulk to the air that had been expelled; then he took it out and weighed it with the water, which gave the weight of the same bulk of water; and on comparing this with the former difference, or weight of air expelled, he found their proportion to be as 1300 to 1. Which is as wide of the truth as Galileo's proportion, namely 400 to 1, but the contrary way. And it is remarkable that the mean between the two, namely 850 to 1, is very near the true proportion as settled by other more accurate experiments.

Mr. Boyle, by a more accurate experiment, found the proportion to be that of 938 to 1. And Mr. Hauksbee found it as 850 to 1, proceeding on the same principles as Mersenne, with a three-gallon glass bottle, but extracting the air out of it with the air pump, instead of expelling it by fire; the height of the barometer being at that time 29.7 inches. Also by other accurate experiments made before the Royal Society by Mr. Hauksbee, Dr. Halley, Mr. Cotes, and others, the proportion was always between 800 and 900 to 1, but rather nearer the latter, namely, being first found as 840 to 1, then as 852 to 1, and a third time as 860 to 1; the barometer then standing at 29 3/4 inches, and the weather warm. Mr. Cavendish determines the ratio 800 to 1, the barometer being 29 3/4, and the thermometer at 50°; and Sir George Shuckburgh, by a very accurate experiment, finds it 836 to 1, the barometer being at that time at 29.27, and the thermometer at 51°. And the medium of all these is about 832 or or 833 to 1, when reduced to the pressure of 30 inches of the barometer, and the mean temperature 55° of the thermometer. Upon the whole therefore it may be safely concluded that, when the barometer is at 30 inches, and the thermometer at the mean temperature 55°, the density or gravity of water is to that of air, as 833 1/3 to 1, that is as 2500/3 to 1, or as 2500 to 3; and that for any changes in the height of the barometer, the ratio varies proportionally; and also that the density of the air is altered by the (1/440)th part for every degree of the thermometer above or below temperate.

This number, which is a very good medium among them all, I have chosen with the fraction 1/37, because it gives exactly 1 1/5 ounce for the mean weight of a cubis soot of air, the weight of the cubic foot of water being just 1000 ounces averdupois, and that of quicksilver equal to 13600 ounces.

Air, then, having been shewn to be a heavy fluid substance, the laws of its gravitation and pressure must be the same as those of water and other fluids; and consequently its pressure must be proportional to its perpendicular altitude. Which is exactly conformable to experiment; for on removing the Terricellian tube to different heights, where the column of air is shorter, the column of quicksilver which it sustains is shorter also, and that nearly at the rate of 100 feet for 1/10 of an inch of quicksilver. And on these principles depend the structure and use of the barometer.

And from the same principle it likewise follows that air, like other fluids, presses equally in all directions. And hence it happens that soft bodies endure this pressure without change of figure, and hard or brittle bodies without breaking; being equally pressed on all parts; but if the pressure be taken off, or diminished, on one side, the effect of it is immediately perceived on the other. See Atmosphere, for the total quantity of effects and pressure, and the laws of different altitudes, &c.

From the weight and fluidity of the air, jointly considered, many effects and uses of it may easily be deduced. By the combination of these two qualities, it closely invests the earth, with all the bodies upon it, constringing and binding them down with a great force, namely a pressure equal to about 15 pounds upon every square inch. Hence, for example, it prevents the arterial vessels of plants and animals from being too much distended by the impetus of the circulating juices, or by the elastic force of the air so copiously abounding in them. For hence it happens, that on a diminution of the pressure of the air, in the operation of cupping, we see the parts of the body grow tumid, which causes an alteration in the circulation of the fluids in the capillary vessels. And the same cause hinders the fluids from transpiring through the pores of their containing vessels, which would otherwise cause the greatest debility, and often destroy the animal. To the same two qualities of the air, weight and fluidity, is owing the mixture of bodies contiguous to one another, especially fluids; for several liquids, as oils and salts, which readily mix of themselves in air, will not mix at all in vacuo. With many other natural phenomena.

3. Elasticity. Another quality of the air, from whence arise a multitude of effects, is its elasticity; a quality by which it yields to the pression of any other bodies, by contracting its volume; and dilates and expands itself again on the removal or diminution of the pressure. This quality is the chief distinctive property of air, the other two being common to other fluids also.

Of this property we have innumerable instances. Thus, for example, a blown bladder being squeezed in the hand, we find a sensible resistance from the included air; and upon taking off the piessure, the compressed parts immediately restore themselves to their former round sigure. And on this property of elasticity depend the structure and uses of the air-pump.

Every particle of air makes a continual effort to dilate itself, and so it acts forcibly against all the neigh- bouring particles, which also exert the like force in return; but if their resistance happen to cease, or be weakened, the particle immediately expands to an immense extent. Hence it is that thin glass bubbles, or bladders, filled with air, and placed under the receiver of an air-pump, do, upon pumping out the air, burst asunder by the force of the air which they contain. So likewise a close flaccid bladder, containing only a small quantity of air, being put under the receiver, swells as the receiver is exhausted, and at length appears quite full. And the same thing happens by carrying the flaccid bladder to the top of a very high mountain.

The same experiment shews that this elastic property of the air is very different from the elasticity of solid bodies, and that these are dilated after a different manner from the air. For when air ceases to be compressed, it not only dilates, but then occupies a far greater space, and exists under a volume immensely greater than before; whereas solid elastic bodies only resume the figure they had before they were compressed.

It is plain that the weight or pressure of the air does not at all depend on its elasticity, and that it is neither more nor less heavy than if it were not at all elastic. But from its being elastic, it follows that it is susceptible of a pressure, which reduces it to such a space, that the force of its elasticity, which re-acts against the pressing weight, is exactly equal to that weight. Now the law os the elasticity is such, that it increases in proportion to the density of the air, and that its density increases in proportion to the forces or weights which compress it. But there is a necessary equality between action and re-action; that is, the gravity of the air, which effects its compression, and the elasticity of it, which gives it its tendency to expansion, are equal.

So that, the elasticity increasing or diminishing, in the same proportion as the density increases or diminishes, that is, as the distance between its particles decrease or increase; it is no matter whether the air be compressed, and retained in any space, by the weight of the atmosphere, or by any other cause; as in either case it must endeavour to expand with the same force. And therefore, if such air as is near the earth be inclosed in a vessel, so as to have no communication with the external air, the pressure of such inclosed air will be exactly equal to that of the whole external atmosphere. And accordingly we find that quicksilver is sustained to the same height, by the elastic force of air inclosed in a glass vessel, as by the whole pressure of the atmosphere.—And on this principle of the condensation and elasticity of the air, depends the structure and use of the air-gun.

That the density of the air is always directly proportional to the force or weight which compresses it, was proved by Boyle and Mariotte, at least as far as their experiments go on this head: and Mr. Mariotte has shewn that the same rule takes place in condensed air. However, this rule is not to be admitted as scrupulously exact; for when air is very forcibly compressed, so as to be reduced to (1/4)th of its ordinary bulk, the effect does not answer precisely to the rule; for in this case the air begins to make a greater resistance, and requires a stronger compression, than according to the rule. And hence it would seem, that the particles of air cannot, by means of any possible weight or pressure, how great soever, be brought into perfect contact, or that it cannot thus be reduced to a solid mass; and consequently that there must be a limit to which this con densation of the air can never arrive. And the same remark is true with regard to the rarefaction of air, namely, that in very high degrees of rarefaction, the elasticity is decreased rather more than in proportion to the weight or density of the air: and hence there must also be a limit to the rarefaction and expansion of the air, by which it is prevented from expanding to infinity.

We know not however how to assign those limits to the elasticity of the air, nor to destroy or alter it, without changing the very nature of air, which is effected by chemical processes. To what degree air is susceptible of condensation, by compression, is not certainly known. Mr. Boyle condensed it 13 times more than in its natural state, by this means: others have compressed it into (1/70)th part of its ordinary volume; Dr. Hales made it 38 times more dense, by means of a press; but by freezing water in a hollow cast-iron ball or shell, he reduced it to 1838 times less space than it naturally occupies; in which state it must have been of more than twice the density or specific gravity of water: And as water is not compressible, except in a very small degree, it follows from this experiment, that the particles of air must be of a nature very different from those of water; since it would otherwise be impossible to reduce air to a volume above 800 times less than in its common state; an inference however which militates directly against an assertion made by Dr. Halley, from some experiments performed in London, and others at Florence by the Academy del Cimento, namely, that it may be safely concluded that no force whatever is capable to reduce air into a space 800 times less than that which it naturally occupies near the surface of the earth.

The elasticity of the air exerts its force equally in all directions; and when it is at liberty, and freed from the cause which compressed it, it expands equally in all directions, and in consequence always assumes a spherical figure in the interstices of the fluids in which it is lodged. This is evident in liquors placed in the receiver of an air pump, by exhausting the air; at first there appears a multitude of exceeding small bubbles, like grains of fine sand, dispersed through the fluid mass, and rising upwards; and as more air is pumped out, they enlarge in size; but still they continue round. Also if a plate of metal be immerged in the liquor, on pumping, its surface will be seen covered over with small round bubbles, composed of the air which adhered to it, now expanding itself. And for the same reason it is that large glass globes are always blown up of a spherical shape, by blowing air through an iron tube into a piece of melted glass at the end of the pipe.

The expansion of the air, by virtue of its elastic property, when only the compressing force is taken off, or diminished, is found to be surprisingly great; and yet we are far from knowing the utmost dilatation of which it is capable. In several experiments made by Mr. Boyle, it expanded first into 9 times its former space; then into 31 times; then into 60, and then into 150 times. Afterwards, it was brought to dilate into 8000 times its first space; then into 10000, and at last even into 13679 times its space; and this solely by its own natural expansive force, by only removing the pressure, but without the help of fire. And on this principle depends the construction and use of the MANOMETER.

The elasticity of the air, under one and the same pressure, is still farther increased by heat, and diminished by cold, and that, by some late accurate experiments made by Sir George Shuckburgh, at the rate of the 440th part of its volume nearly, for each degree of the variation of heat, from that of temperate, in Fahrenheit's thermometer.

Mr. Hauksbee observed that a portion of air inclosed in a glass tube, when the temperature was at the freezing point, formed a volume which was to that of the same quantity of air in the greatest heat of summer here in England, as 6 to 7. And it has been found by several experiments, that air is expanded 1/3 of its natural bulk by applying the heat of boiling water to it.

Dr. Hales found that the air in a retort, when the bottom of the vessel just became red hot, was dilated into twice its former space; and that in a white, or almost melting heat, it filled thrice its former space: but Mr. Robins found that air was expanded, by means of the white or fusing heat of iron, to 4 times its former bulk.

See several ingenious experiments on the elasticity of the air, in the Philos. Trans. for the year 1777, by Sir George Shuckburgh and Colonel Roy.

This properly explains the common effect observed on bringing a close flaccid bladder near the fire to warm it; when it is presently found to swell as if more air were blown into it. And upon this principle depends the structure and office of the thermometer; as also the air balloons, lately invented by Mr. Montgolfier, for floating in the atmosphere.

M. Amontons first discovered that, with the same degree of heat, air will expand in a degree proportioned to its density. And on this foundation the ingenious author has formed a discourse, to prove “that the spring and weight of the air, with a moderate degree of warmth, may enable it to produce even earthquakes, and others of the most vehement commotions of nature.” He computes that at the depth of the 74th part of the earth's radius below the surface, the natural pressure of the air would reduce to the density of gold; and thence infers that all matter below that depth, is probably heavier than the heaviest metal that we know of. And hence again, as it is proved that the more the air is compressed, the more does the same degree of fire increase the force of its elasticity; we may infer that a degree of heat, which in our orb can produee only a moderate effect, may have a very violent one in such lower orb; and that, as there are many degrees of heat in nature, beyond that of boiling water, it is probable there may be some whose violence, thus assisted by the weight of the air, may be sufficiently powerful to tear asunder the solid globe. Mem. de l'Acad. 1703.

Many philosophers have supposed that the elastic property of the air depends on the figure of its corpuscles, which they take to be ramous: some maintain that they are so many minute flocculi, resembling fleeces of wool: others conceive them rolled up like hoops, and curled like wires, or shavings of wood, or coiled like the springs of watches, and endeavouring to expand themselves by virtue of their texture.

But Sir Isaac Newton (Optics, Qu. 31, &c.) explains the matter in a different way; such a contexture of parts he thinks by no means sufficient to account for that amazing power of elasticity observed in air, which is capable of dilating itself into above a million of times more space than it occupied before: but, he observes, as it is known that all bodies have an attractive and a repelling power; and as both these are stronger in bodies, the denser, more compact, and solid they are; hence it follows that when, by heat, or any other powerful agent, the attractive force is overcome, and the particles of the body separated so far as to be out of the sphere of attraction; the repelling power, then commencing, makes them recede from each other with a strong force, proportionable to that with which they before cohered; and thus they become permanent air.

And hence, he says, it is, that as the particles of air are grosser, and rise from denser bodies, than those of transient air, or vapour, true air is more ponderous than vapour, and a moist atmosphere lighter than a dry one.

And M. Amontons makes the elasticity of air to arise from the fire it contains; so that by augmenting the degree of heat, the rarefaction will be increased to a far greater degree than by a mere spontaneous dilatation.

The elastic power of the air becomes the second great source of the remarkable effects of this important fluid. By this property it insinuates itself into the pores of bodies, where, by means of this virtue of expanding, which is so easily excited, it must put the particles of those bodies into perpetual vibrations, and maintain a continual motion of dilatation and contraction in all bodies, by the incessant changes in its gravity and density, and consequently its elasticity and expansion.

This reciprocation is observable in several instances, particularly in plants, in which the tracheæ or air-vessels perform the office of lungs; for as the heat increases or diminishes, the air alternately dilates and contracts, and so by turns compresses the vessels, and eases them again; thus promoting a circulation of their juices. And hence it is found that no vegetation or germination is carried on in vacuo.

It is from the same cause too, that ice is burst by the continual action of the air contained in its bubbles. Thus, too, glasses and other vessels are frequently cracked, when their contained liquors are frozen; and thus also large blocks of stone, and entire columns of marble, sometimes split in the winter season, from some little bubble of included air acquiring an increased elasticity: and for the same reason it is that so few stones will bear to be heated by a fire, without cracking into many pieces, by the increased expansive force of some air confined within their pores. From the same source arise also all putrefaction and fermentation; neither of which can be carried on in vacuo, even in the best disposed subjects. And even respiration, and animal-life itself, are supposed, by many authors, to be conducted, in a great measure, by the same principle of the air. And as we find such great quantities of air generated by the solution of animal and vegetable substances, a good deal must constantly be raised from the dissolution of these clements in the stomach and bowels.

In fact, all natural corruption and alteration seem to depend on air; and even metals, particularly gold, only seem to be durable and incorruptible, in so far as they are impervious to air.

As to the different kinds of air, with its generation, and the effects of different ingredients of it, &c, they are omitted here, as properly belonging to a Chemical Dictionary, or to a General Dictionary of Arts, &c.

For the resistance of the air, see Resistance.

Air-Gun, in Pneumatics, is a machine for propelling bulleto with great violence, by the sole means of condensed air.

The first account we meet with of an air-gun, is in the Elemens d'Artillerie of David Rivaut, who was preceptor to Louis XIII. of France. He ascribes the invention to one Marin, a burgher of Lisieux, who presented one to Henry IV.

To construct a machine of this kind, it is only necessary to take a strong vessel of any sort, into which the air is to be thrown or condensed by means of a syringe, or otherwise, the more the better; then a valve is suddenly opened, which lets the air escape by a small tube in which a bullet is placed, and which is thus violently forced out before the air.

It is evident then that the effect is produced by virtue of the elastic property of the air; the force of which, as has been shewn in the last article, is directly proportional to its condensation; and therefore the greater quantity that can be forced into the engine, the greater will be the effect. Now this effect will be exactly similar to that of a gun charged with powder, and therefore we can easily form a comparison between them: for inflamed gun-powder is nothing more than very condensed elastic air; so that the two forces are exactly similar. Now it is shewn by Mr. Robins, in his New Principles of Gunnery, that the fluid of inflamed gun-powder, has, at the first moment, a force of elasticity equal to about a 1000 times that of common air; and therefore it is necessary that air should be condensed a 1000 times more than in its natural state, to produce the same effect as gun-powder. But then it is to be considered, that the velocities with which equal balls are impelled, are directly proportional to the square roots of the forces; so that if the air in an air-gun be condensed only 10 times, then the velocity it will project a ball with, will be, by that rule, (1/10)th of that arising from gun-powder; and if the air were condensed 20 times, it would communicate a velocity of 1/7 of that of gun-powder. But in reality the air-gun shoots its ball with a much greater proportion of velocity than as above, and for this reason, namely, that as the reservoir, or magazine of condensed air, is commonly very large in proportion to the tube which contains the ball, its density is very little altered by expanding through that narrow tube, and consequently the ball is urged all the way by nearly the same uniform force as at the first instant; whereas the elastic fluid arising from inflamed gun-powder is but very small in proportion to the tube or barrel of the gun, occupying at first indeed but a very small portion of it next the but-end: an dtherefore by dilating into a comparatively large space, as it urges the ball along the barrel, its elastic force is proportionally weakened, and it acts always less and less on the ball in the tube. From which cause it happens, that air condensed into a good large machine only 10 times, will shoot its ball with a velocity but little inferior to that given by the gunpowder. And if the valve of communication be suddenly shut again by a spring, after opening it to let some air escape, then the same collection of it may serve to impel many balls, one after another.

In all cases in which a considerable force is required, and consequently a great condensation of air, it will be requisite to have the condensing syringe of a small bore, perhaps not more than half an inch in diameter: otherwise the force to produce the compression will become so great, that the operator cannot work the machine: for, as the pressure against every square inch is about 15 pounds, and against every circular inch about 12 pounds, if the syringe be one inch in diameter, when one atmosphere is injected, there will be a resistance of 12 pounds against the piston; when 2, of 24 pounds; and when 10 are injected, there will be a force of 120 pounds to overcome; whereas 10 atmospheres act against the half-inch piston, whose area is but 1/4 of the former, with 1/4 of the force only, namely, 30 pounds; and 40 at mospheres may be injected with such a syringe, as well as ten with the larger.

There are air-guns of various constructions; an easy and portable one is represented in Plate II, fig. 1. which is a section lengthways through the axis, to shew the inside. It is made ofbrass, and has two barrels; the inner barrel D A of a small bore, from which the bullets are shot; and a larger barrel ESCDR, on the outside of it. In the stock of the gun there is a syringe MNPS, whose rod M draws out to take in air; and by pushing it in again, the piston SN drives the air before it, through the valve PE into the cavity between the two barrels. The ball K is put down into its place in the small barrel, with the rammer, as in another gun. There is another valve at SL, which, being opened by the trigger O, permits the air to come behind the ball, so as to drive it out with great force. If this valve be opened and shut suddenly, one charge of condensed air may make several discharges of bullets; because only part of the injected air will then go out at a time, and another bullet may be put into the place K: but if the whole air be discharged on a single bullet, it will impel it more forcibly. This discharge is effected by means of a lock (fig. 2) when fixed to its place as usual in other guns; for the trigger being pulled, the cock will go down and drive a lever which opens the valve.

Dr. Macbride (Exper. Ess. p. 81) mentions an improvement of the air-gun, made by Dr. Ellis; in which the chamber sor containing the condensed air is not in the stock, which renders the machine heavy and unweildy, but has five or six hollow spheres belonging to it, of about 3 inches diameter, sitted to a screw on the lock of the gun. These spheres are contrived with valves, to consine the air which is forced into their cavities, so that a servant may carry them ready charged with condensed air: and thus the gun of this construction is rendered as light and portable as one of the smallest fowling.pieces.

Fig. 3 represents one made by the late Mr. B. Martin of London, and now by several of the mathematical instrument and gun-makers of the metropolis; which, for simplicity and perfection, perhaps exceeds any other that has been contrived. A is the gun-barrel, of the size and weight of a common fowling-piece, with the lock, stock, and ramrod. Under the lock, at b, is a round steel tube, having a small moveable pin in the inside, which is pushed out when the trigger a is pulled, by the springwork within the lock; to this tube b is serewed a hollow copper ball, perfectly airtight. This copper ball is fully charged with condensed air by means of a syringe, previous to its being applied to the tube b. Hence, if a bullet be rammed down in the barrel, the copper ball screwed fast at b, and the trigger a be pulled; then the pin in b will forcibly push open a valve within the copper ball, and let out a portion of the condensed air; which air will rush up through the aperture of the lock, and forcibly act against the bullet, driving it to the distance of 60 or 70 yards, or farther. If the air be strongly condensed at every discharge, only a portion of the air escapes from the ball; therefore, by re-cocking the piece, another discharge may be made; and this repeated 15 or 16 times. An additional barrel is sometimes made, and applied for the discharge of shot, instead of the ball above described.

Sometimes the syringe is applied to the end of the barrel C (fig. 4); the lock and trigger shut up in a brass case d; and the trigger pulled, or the discharge made, by pulling the chain b. In this contrivance there is a round chamber for the condensed air at the end of the spring at e, and it has a valve acting in a similar manner to that of the copper ball. When this instrument is not in use, the brass case d is made to slide off, and the instrument then becomes a walking stick: from which circumstance, and the barrel being made of cane, or brass, &c, it has been called the Air-cane. The head of the cane unscrews and takes off at a, where the extremity of the piston-rod in the barrel is shewn. An iron rod is placed in a ring at the end of this, and the air is condensed in the barrel in a manner similar to that of the gun as above; but its force and action is not near so strong as in the gun.

Magazine Air-Gun. This is an improvement of the common air-gun, made by an ingenious artist, called L. Colbe. By his contrivance, ten bullets are so lodged in a cavity, near the place of discharge, that they may be successively drawn into the barrel, and shot so quickly as to be nearly of the same use as so many different guns; the only motion required, after the air has been injected, being that of shutting and opening the hammer, and cocking and pulling the trigger. Fig. 3 is a longitudinal section of this gun, as large in every part as the gun itself; and as much of its length is shewn as is peculiar to this construction; the rest of it being like the ordinary air-gun. EE is part of the stock; G is the end of the injecting syringe, with its valve H, opening into the cavity FFFF between the barrels, KK is the small or shooting barrel, which receives the bullets, one at a time, from the magazine DE, being a serpentine cavity, in which the bullets b, b, b, &c, are lodged, and closed at the end D; from whence, by one motion of the hammer, they are brought into the barrel at I, and thence are shot out by the opening of the valve V, which lets in the condensed air from the cavity FFF into the channel VKI, and so along the inner barrel KKK, whence the bullet is discharged. s I si M k is the key of a cook, having a hole through it; which hole, in the present situation, makes part of the barrel KK, being just of the same bore: so that the air, which is let in at every opening of the valve V, comes behind this cock, and taking the ball out of it, carries it forward, and so out of the mouth of the piece.

To bring in another bullet to succeed I, which is done in an instant, bring the cylindrical cavity of the key of the cock, which made part of the barrel KKK, into the situation ik, so that the part I may be at K; then turning the gun upside-down, one bullet next the cock will fall into it out of the magazine, but will go no farther into this cylindrical cavity, than the two little pieces ss will permit it; by which means only one bullet at a time will be taken in to the place I, to be discharged again as before.

A more particular description of the several parts may be seen in Desaguliers' Exper. Philos. vol. ii. pa. 399 et seq.

Air-Pump, in Pneumatics, is a machine for exhausting the air out of a proper vessel, and so to make what is commonly called a vacuum; though in reality the air in the receiver is only rarefied to a great degree, so as to take off the ordinary effects of the atmosphere. So that by this machine we learn, in some measure, what our earth would be without air; and how much all vital, generative, nutritive, and alterative powers depend upon it.

The principle on which the air-pump is constructed, is the spring or elasticity of the air; as that on which the common, or water pump is formed, is the gravity of the same air: the one gradually exhausting the air from a vessel by means of a piston, with a proper valve, working in a cylindrical barrel or tube; and the other exhausting water in a similar manner.

The air-pump has proved one of the principal means of performing philosophical discoveries, that has been invented by the moderns. The idea of such a machine occurred to several persons, nearly about the same time. But the first it seems was completed by Otto Guericke, the celebrated consul of Magdeburg, who exhibited his first public experiments with it, before the emperor and the states of Germany, at the breaking up of the imperial diet at Ratisbon, in the year 1654. But it was not till the year 1672 that Guericke published a description of the instrument, with an account of his experiments, in his Experimenta Nova Magdeburgica de Vacuo Spacia: though an account of them had been published by Schottus in 1657, in his Mechanica Hydraulico Pneumatica.

Dr. Hook and M. Duhamel ascribe the invention of the air-pump to Mr. Boyle. But that great man frankly confesses that Guericke was beforehand with him in the execution. Some attempts, he assures us, he had indeed made upon the same foundation, before he knew any thing of what had been done abroad: but the information he afterwards received from the account given by Schottus, enabled him, with the assistance of Dr. Hook, aster two or three unsuccessful trials, to bring his design to maturity. The product of their labours was a new air-pump, much more easy; convenient, and manageable, than the German one. And hence, or rather from the great variety of experiments to which this illustrious author applied the machine, it was afterwards called Machina Boyliana, and the vacuum produced by it, Vacuum Boylianum.

Structure of the Air-Pump. Most of the air-pumps that were first made, consisted of only one barrel, or hollow cylinder of brass, with a valve at the bottom, opening inwards; and a moveable embolus or piston, having likewise a valve opening upwards, and so exactly fitted to the barrel, that when it is drawn up from the bottom, by means of an indented iron rod or rack, and a handle turning a small indented wheel, playing in the teeth of that rod, all the air will be drawn up from the cavity of the barrel: there is also a small pipe opening into the bottom of the barrel, by means of which it communicates with any proper vessel to be exhausted of air, which is called a receiver, from its office in receiving the subjects upon which experiments are to be made in vacuo: the whole being fixed in a convenient frame of wood-work, where the end of the pipe turns up into a horizontal plate, upon which the receiver is placed, just over that end of the pipe.

The other parts of the machine, being only accidental circumstances, chiefly respecting conveniency, have been diversified and improved from time to time, according to the address and several views of the makers. That of Otto Guericke was very rude and inconvenient, requiring the labour of two strong men, for more than two hours, to extract the air from a glass, which was also placed under water; and yet allowed of no change of subjects for experiments.

Mr. Boyle, from time to time, removed several of these inconveniences, and lessened others: but still the working of his pump, which had but one barrel, was laborious, by reason of the pressure of the atmosphere, a great part of which was to be removed at every lift of the piston, when the exhaustion was nearly completed. Various improvements were successively made in the machine by the philosophers about that time, and foon after, who cultivated this new and important branch of pneumatics; as Papin, Mersenne, Mariotte, and others; but still they laboured under a difficulty of working them, from the circumstance of the single barrel, till Papin, in his farther improvements of the air-pump, removed that inconvenience, by the use of a second barrel and piston, contrived to rise, as the other fell, and to sall as that rose; by which, and the great improvements made by Mr. Hauksbee, the pressure of the atmosphere on the descending piston, always nearly balanced that of the ascending one; so that the winch, which worked them up and down, was easily moved by a very gentle force with one hand; and besides, the exhaustion was hereby made in less than half the time.

Some of the Germans, and others likewise, made improvements in the air-pump, and contrived it to perform the counter office of a condenser, in order to examine the properties of the air depending on its condensation.

Mr. Boyle contrived a mercurial gauge or index to the air-pump, which is described in his first and second Physico-Mechanical Continuations, for measuring the degrees of the air's rarefaction in the receiver. This gauge is similar to the barometer, being a long glass tube, having its lower end immersed in an open bason of quicksilver, but its other end, which was open also, communicating with the receiver: which being exhausted, this tube is equally exhausted of air at the same time, and the external air presses the quicksilver up into the tube, to a height proportioned to the degree of exhaustion.

Mr. Vream, an ingenious pneumatic operator, made an improvement in Hauksbee's air-pump, by reducing the alternate up-and-down motion of the hand and winch to a circular one. In his method, the winch is turned quite round, and yet the pistons are alternately raised and depressed: by which the trouble of shifting the hand backwards and forwards, as well as the loss of time, and the shaking of the pump, are prevented.

The air-pump, thus improved, is represented in plate III. fig. 1; where oo is the receiver to be exhausted, ground truly level at the bottom, set over a hole in the plate, from which descends the bent pipe hh to the cistern dd, with which the two barrels aa communicate, in which the pistons are worked by a toothed wheel, by turning the handle bb; by which the racks cc, with the pistons, are worked alternately up and down. ll is the gauge tube, immersed in a bason of quicksilver m at bottom, and communicating with the receiver at top; from which however it may be occasionally disengaged, by turning a cock. And n is another cock, by turning of which, the air is again let in to the exhausted receiver; into which it is heard to rush with a considerable hissing noise.

Notwithstanding the great excellency of Mr. Hauksbee's air-pump, it was still subject to inconveniences, from which it was in a great measure relieved by some contrivances of Mr. Smeaton, which are described at large in the Philos. Trans. for the year 1752. The principal improvements suggested by Mr. Smeaton, relate to the gauge, the valves of the piston, and the piston going closer down to the bottom of the barrel; for his pump has only one. By the last of these, the air was extracted more perfectly at each stroke. By the second, he remedied an inconvenience arising from the valve hole of the piston being too wide properly to support the bladder valve which covered it: instead of the usual circular orisice, Mr. Smeaton perforated the piston with seven small and equal hexagonal holes, one in the centre, and the other six around, forming together the appearance of a transverse section of a honeycomb; the bars or divisions between which, served to support the pressure of the air on the valve. His gage consists of a bulb of glass, of a pear-like shape, and capable of holding about half a pound of quicksilver: it is open at the lower end, the other terminating in a tube hermetically sealed; and it has annexed to it a seale, divided into parts of about 1/10 of an inch, and answering to the 1000th part of the whole capacity. During the exhaustion of the receiver, the gage is suspended in it by a wire; but when the pump has been worked as much as necessary, the gage is pushed down, till the open end be immersed in a bason of quicksilver placed underneath. The air is then let into the receiver again, and the quicksilver driven by it from the bason, up into the gauge, till the air remaining in it become of the same density as the air without; and as the air always takes the highest place, the tube being uppermost, the expansion will be determined by the number of divisions occupied by the air at the top. This airpump is made to act also as a condensing engine, as some German machines had done before, by the very simple apparatus of turning a cock.

By means of this gauge, Mr. Smeaton judged that his machine was incomparably better than any former ones, as it seemed to rarefy the air in the receiver 1000, or even 2000 times, while the best of the former construction only rarefied about 140 times: and so the case has since been always understood, an implicit considence being placed in Mr. Smeaton's accuracy, till the fallacy was accidentally detected in the manner related at large by Mr. Nairne in the Philos. Trans. for the year 1777. This accurate and ingenious artist wanting to make trial of Mr. Smeaton's pear-gauge, executed an air-pump of his improved construction, in the best manner possible; which, in various experiments made with it, appeared, by the pear-gauge, to rarefy the air to an amazing degree indeed, being at times from 4000 to 10000, or 50000, or even 100000 times rarefied. But upon measuring the same expansion by the usual long and short tube gauges, which both accurately agreed together, he found that these never shewed a rarefaction of more than 600 times: widely different from the same as measured by the pear or internal gauge, by experiments often repeated. ‘Finding, says Mr. Nairne, still this disagreement between the pear-gauge and the other gauges, I tried a variety of experiments; but none of them appeared to me satisfactory, till one day in April 1776, shewing an experiment with one of these pumps to the honourable Henry Cavendish, Mr. Smeaton, and several other gentlemen of the Royal Society, when the two gauges differed some thousand times from one another, Mr. Cavendish accounted for it in the following manner. “It appeared, he said, from some experiments of his father's, Lord Cavendish, that water, whenever the pressure of the atmosphere on it is diminished to a certain degree, is immediately turned into vapour, and is as immediately turned back again into water on reftoring the pressure. This degree of pressure is different according to the heat of the water: when the heat is 72° of Fahrenhest's scale, it turns into vapour as soon as the pressure is no greater than that of three quarters of an inch of quicksilver, or about 1-40th of the usual pressure of the atmosphere; but when the heat is only 41°, the pressure must be reduced to that of a quarter of an inch of quicksilver before the water turns into vapour. It is true, that water exposed to the open air, will evaporate at any heat, and with any pressure of the atmosphere; but that evaporation is intirely owing to the action of the air upon it; whereas the evaporation here spoken of, is performed without any assistance from the air. Hence it follows, that when the receiver is exhausted to the above-mentioned degree, the moisture adhering to the different parts of the machine will turn into vapour, and supply the place of the air, which is continually drawn away by the working of the pump; so that the fluid in the pear-gauge, as well as that in the receiver, will consist in a good measure of vapour. Now letting the air into the receiver, all the vapour within the pear-gauge will be reduced to water, and only the real air will remain uncondensed; consequently the pear-gauge shews only how much real air is left in the receiver, and not how much the pressure or spring of the included fluid is diminished; whereas the common gauges shew how much the pressure of the included fluid is diminished, and that equally, whether it consist of air or of vapour.” Mr. Cavendish having explained so satisfactorily the cause of the disagreement between the two gauges, Mr. Nairne considered that, if he were to avoid moisture as much as possible, the two gauges should nearly agree. And in fact they were found so to do, each shewing a rarefaction of about 600, when all moisture was perfectly cleared away from the pump, and the plate and the edges of the receiver were secured by a cement instead of setting it upon a soaked leather, as in the usual way. But by future experiments, Mr. Nairne found that the same excellent machine would not exhaust more than 50 or 60 times, when the receiver was set upon leather soaked in water, the heat of the room being about 57°. And from the whole, Mr. Nairne concludes that the air-pump of Otto Guericke, and those contrived by Mr. Gratorix, and Dr. Hook, and the improved one by Mr. Papin, both used by Mr. Boyle, as also Hauksbee's, s'Gravesande's, Muschenbroeck's, and those of all who have used water in the barrels of their pumps, could never have exhausted to more than between 40 and 50, if the heat of the place was about 57; and although Mr. Smeaton, with his pump, where no water was in the barrel, but where leather soaked in a mixture of water and spirit of wine was used on the pump-plate, to set the receiver upon, may have exhausted all but a thousandth, or even a tenthousandth part of the common air, according to the testimony of his pear-gauge; yet so much vapour must have arisen from the wet leather, that the contents of the receiver could never be less than a 70th or 80th part of the density of the atmosphere. But when nothing of moisture is used about this machine, it will, when in its greatest perfection, rarefy its contents of air about 600 times.

It is evident that by means of these two gauges we can ascertain the several quantities of vapour and permanent air which make up the contents of the receiver, after the exhaustion is made as perfect as can be; for the usual external gauge determines the whole contents, made up of the vapour and air, whilst the pear-gauge shews the quantity of real permanent air; consequently the difference is the quantity of vapour.

The principal cause which prevents this pump from exhausting beyond the limit above-mentioned, is the weakened elasticity of the air within the receiver, which, decreasing in proportion as the quantity of the air within is diminished, becomes at last incapable of lifting up the valve of communication between the receiver and the barrel; and consequently no more air can then pass from the former to the latter.

Several ingenious persons have used their endeavours to remove this imperfection in the best air-pumps. Amongst these it seems that one Mr. Haas has succeeded tolerably well; having, by means of a contri- vance to open the communication valve in the bottom of the barrel, made his machine so perfect, that when every thing is in the greatest perfection, it rarefies the contents of the receiver as far as 1000 times, even when measured by the exterior gauge. The description of this machine, and an account of some experiments performed with it, are given by Mr. Cavallo in the Philos. Trans. for the year 1783.

But the imperfections it seems have more recently been removed by an ingenious contrivance of Mr. Cuthbertson, a mathematical instrument maker at Amsterdam, now of London, whose air-pump has neither cocks nor valves, and is so constructed, that what supplies their place has the advantages of both, without the inconveniences of either. He has also made improvements in the gauges, by means of which he determines the height of the mercury in the tube, by which the degree of exhaustion is indicated, to the hundredth part of an inch. And to obviate the inconvenience of the elastic vapour arising from the wet leather, upon which the receiver is placed, for common experiments, he recommends the use of leather dressed with allum, and soaked in hog's lard, which he found to yield very little of this vapour; but when the utmost degree of exhaustion is required, his advice is, to dry the receiver well, and set it upon the plate without any leather, only smearing its outer edges with hog's lard, or with a mixture of three parts of hog's lard and one of oil. But the use of the leather has long been laid aside by our English instrument-makers, a circumstance which probably had not come to Mr. Cuthbertson's knowledge. An account of this instrument, and of some experiments performed with it, was published at Amsterdam in the year 1787; from which experiments it appears that, by a coincidence of the several gauges, a rarefaction of 1200 times was shewn; but when the atmosphere was very dry, the exhaustion has been so complete, that the gauges have shewn the air in the receiver to be rarefied above 2400 times.

There are made also by different persons, portable, or small air-pumps, of various constructions, to set upon a table, to perform experiments with. In these, the gauge is varied according to the fancy of the maker, but commonly it consists of a bent glass tube, like a syphon, open only at one end. The gauge is placed under a small receiver communicating, by a pipe, with the principal pipe leading from the general receiver to the barrels. The close end of the gauge, of 3 or 4 inches long, before the exhaustion, has the quicksilver forced close up to the top by the pressure of the air on the open end; but when the exhaustion is considerably advanced, it begins to descend, and then the difference of the heights of the quicksilver in the two legs, compared with the height in the barometrical tube, determines the degree of exhaustion: so if the difference between the two be one inch, when the barometer stands at 30, the air is rarefied 30 times; but if the difference be only half an inch, the rarefaction is 60 times, and so on. See Plate 111. fig. 2.

The Use of the Air-Pump. In whatever manner or form this machine be made, the use and operation of it are always the same. The handle, which works the piston, is moved up and down in the barrel, by which means a barrel of the contained air is drawn out at every stroke of the piston, in the following manner: by pushing the piston down to the bottom of the barrel, where the air is prevented from escaping downwards, by its elasticity it opens the valve of the piston, and escapes upwards above it into the open air; then raising the piston up, the external atmosphere shuts down its valve, and a vacuum would be made below it, but for the air in the receiver, pipe, &c, which now raises the valve in the bottom of the barrel, and rushes in and fills it again, till the whole air in the receiver and barrel be of one uniform density, but less than it was before the stroke, in proportion as the sum of all the capacities of the receiver, pipe, and barrel together, is to the same sum wanting the barrel. And thus is the air in the receiver diminished at each stroke of the piston, by the quantity of the barrel or cylinder full, and therefore always in the same proportion: so that by thus repeating the operation again and again, the air is rarefied to any proposed degree, or till it has not elasticity enough to open the valve of the piston or of the barrel, after which the exhaustion cannot be any farther carried on: the gauge, in comparison with the barometer, shewing at any time what the degree of exhaustion is, according to the particular nature and construction of it.

But, supposing no vapour from moisture, &c, to rise in the receiver, the degree of exhaustion, after any number of strokes of the piston, may be determined by knowing the respective capacities of the barrel and the receiver, including the pipe, &c. For as we have seen above that every stroke diminishes the density in a constant proportion, namely as much as the whole content exceeds that of the cylinder or barrel; and consequently the sum of as many diminutions as there are strokes of the piston, will shew the whole diminution by all the strokes. So, if the capacity of the barrel be equal to that of the receiver, in which the communication pipe is always to be included; then, the barrel being half the sum of the whole contents, half the air will be drawn out at one stroke; and consequently the remaining half, being dilated through the whole or first capacity, will be of only half the density of the first: in like manner, after the second stroke, the density of the remaining contents will be only half of that after the sirst stroke, that is only 1/4 of the original density: continuing this operation, it follows that the density of the remaining air will be 1/8 after 3 strokes of the piston, 1/16 after 4 strokes, 1/32 after 5 strokes, and so on, according to the powers of the ratio 1/2; that is, such power of the ratio as is denoted by the number of the strokes. In like manner, if the barrel be 1/3 of the whole contents, that is, the receiver double of the barrel, or 2/3 of the whole contents; then the ratio of diminution of density being 2/3, the density of the contents, after any number of strokes of the piston, will be denoted by such power of 2/3 whose exponent is that number; namely, the density will be 2/3 after one stroke, (2/3)2 or 4/9 after two strokes, (2/3)3 or 8/27 after 3 strokes, and in general it will be (2/3)n after n strokes: the original density of the air being 1. Hence then, universally, if s denote the sum of the contents of the receiver and barrel, and r that of the receiver only without the barrel, and n any number of strokes of the piston; then, the original density of the air being 1, the density after n strokes will be (r/s)n o rn/sn, namely the n power of the ratio r/s. So, for example, if the capacity of the receiver be equal to 4 times that of the barrel; then their sum s is 5, and r is 4; and the density of the contents after 30 strokes, will be (4/5)30, or the 30th power of 4/5, which is 1/808 nearly; so that the air in the receiver is raresied 808 times.

See also the Memoires de l'Acad. Royale des Sciences for the years 1693 and 1705.

From the same formula, namely (r/s)n = d the density, we easily derive a rule for finding the number of strokes of the piston, necessary to rarefy the air any number of times, or to reduce it to a given density d, that of the natural air being 1. For since ((r/s)n = d, by taking the logarithm of this equation, it is n X log. r/s = log. of d; and hence ; that is, divide the log. of the proposed density by the log. of the ratio of the receiver to the sum of the receiver and barrel together, and the quotient will shew the number of strokes of the piston requisite to produce the degree of exhaustion required. So, for example, if the receiver be equal to 5 times the barrel, and it be proposed to find how many strokes of the piston will rarefy the air 100 times; then r = 5, s = 6, d = 1/100, whose log. is - 2, and r/s=5/6, whose log. is - .07918; therefore 2/.07918 = 25 1/4 nearly, which is the number of strokes required.

And, farther, the same formula reduced, would give us the proportion between the receiver and barrel, when the air is rarefied to any degree by an assigned number of strokes of the piston. For since the density, therefore, extracting the n root of both sides, it is : that is, the n root of the density is equal to the ratio of the receiver to the sum of the receiver and barrel. So, if the density d be 1/128, and the number of strokes n = 7; then the 7th root of 1/128 is 1/2; which shews that the receiver is equal to half the receiver and barrel together, or that the capacity of the barrel is just equal to that of the receiver.

Some of the principal effects and phenomena of the air-pump, are the following: That, in the exhausted receiver, heavy and light bodies fall equally swift; so, a guinea and feather fall from the top of a tall receiver to the bottom exactly together. That most animals die in a minute or two: but however, That vipers and frogs, though they swell much, live an hour or two; and after being seemingly quite dead, come to life again in the open air: That snails survive about ten hours; efts, or slow-worms, two or three days; and leeches five or six. That oysters live for 24 hours. That the heart of an eel taken out of the body, continues to beat for good part of an hour, and that more briskly than in the air. That warm blood, milk, gall, &c, undergo a considerable intumescence and ebullition. That a mouse or other animal may be brought, by degrees, to survive longer in a rarefied air, than naturally it does. That air may retain its usual pressure, after it is become unfit for respiration. That the eggs of silk-worms hatch in vacuo. That vegetation stops. That fire extinguishes; the flame of a candle usually going out in one minute; and a charcoal in about five minutes. That red-hot iron, however, seems not to be affected; and yet sulphur or gun-powder are not lighted by it, but only fused. That a match, after lying seemingly extinct a long time, revives again on re-admitting the air. That a flint and steel strike sparks of fire as copiously, and in all directions, as in air. That magnets, and magnetic needles, act the same as in air. That the smoke of an extinguished luminary gradually settles to the bottom in a darkish body, leaving the upper part of the receiver clear and transparent; and that on inclining the vessel sometimes to one side, and sometimes to another, the fume preserves its surface horizontal, after the nature of other fluids. That heat may be produced by attrition. That camphire will not take fire; and that gun-powder, though some of the grains of a heap of it be kindled by a burning glass, will not give fire to the contiguous grains. That glow-worms lose their light in proportion as the air is exhausted, and at length become totally obscure; but on re-admitting the air, they presently recover it all. That a bell, on being struck, is not heard to ring, or very faintly. That water freezes. But that a syphon will not run. That electricity appears like the aurora borealis. With multitudes of other curious and important particulars, to be met with in the numerous writings on this machine, namely, besides the Philos. Transactions of most academies and societies, in the writings of Torricelli, Pascal, Mersenne, Guericke, Schottus, Boyle, Hook, Duhamel, Mariotte, Hauksbee, Hales, Muschenbroeck, Gravesande, Desaguliers, Franklin, Cotes, Helsham, and a great many other authors.

Air-Vessel, in Hydraulics, is a vessel of air within some water engines, which being compressed, by forcing in a considerable quantity of water, by its uniform spring, forces it out at the pipe in a constant uninterrupted stream, to a great height.

Air-vessel too, in the improved fire engines, is a metallic cylinder, placed between the two forcing pumps, by the action of whose pistons the water is forced into this vessel, through two pipes, with valves; then the air, previously contained in it, is compressed by the water, in proportion to the quantity admitted, and this air, by its spring, forces the water through a pipe by a constant and equal stream; whereas in the common squirting engine, the stream is discontinued between the several strokes.

AIRY Triplicity, in Astrology, the signs of Gemini, Libra, and Aquarius.

, or Adjutage, in Hydraulics, part of the apparatus of a jet d'eau, or artisicial fountain; being a kind of tube fitted to the aperture or mouth of the cistern, or the pipe; through which the water is to be played in any direction, and in any shape or figure.

It is chiefly the diversity in the ajutage, that makes the different kinds of fountains. So that, by having several ajutages, to be applied occasionally, one fountain is made to have the effect of many.

Mariotte, Gravesande, and Desaguliers have written pretty fully on the nature of ajutages, or spouts for jets d'eau, and especially the former. He affirms, from experiment, that an even polished round hole, made in the thin end of a pipe, gives a higher jet than either a cylindrical or a conical ajutage; but that, of these two latter however, the conical is better than the cylindrical figure. See his Traite du Mouvement des Eaux, part 4.

The quantity of water discharged by ajutages of equal area, but of different figures, is the same. And for like figures, but of different sizes, the quantity discharged, is directly proportional to the area of the ajutage, or to the square of its diameter, or of any side or other linear dimension: so, an ajutage of a double diameter, or side, will discharge 4 times the quantity of water; of a triple diameter, 9 times the quantity; and so on; supposing them at an equal depth below the surface or head of water. But if the ajutage be at different depths below the head, then the celerity with which the water issues, and consequently the quantity of it run out in any given-time, is directly proportional to the square-root of the altitude of the head, or depth of the hole: so at 4 times the depth, the celerity and quantity is double; at 9 times the depth, triple; and so on.

It has been found that jets do not rise quite so high as the head of water; owing chiefly to the resistance of the air against it, and the pressure of the upper parts of the jet upon the lower: and for this reason it is, that if the direction of the ajutage be turned a very little from the perpendicular, it is found to spout rather higher than when the jet is exactly upright.

It is sound by experiment too, that the jet is higher or lower, according to the size of the ajutage: that a circular hole of about an inch and a quarter in diameter, jets highest; and that the farther from that size, the worse. Experience also shews that the pipe leading to the ajutage, should be much larger than it; and if the pipe be along one, that it should be wider the farther it is from the ajutage.

For the other circumstances relating to jets and the issuing of water under various circumstances, see EXHAUSTION, Flux, Fountain, Jet d'Eau, &c, to which they more properly belong.

, an Arabic prince of Batan in Mesopotamia, who was a celebrated astronomer, about the year of Christ 880, as appears by his observations. He is also called Muhammed ben Geber Albatani, Mahomet the son of Geber, and Muhamedes Aractensis. He made astronomical observations at Antioch, and at Racah or Aracta, a town os Chaldea, which some authors call a town of Syria or of Mesopotamia. He is highly spoken of by Dr. Halley, as a vir admirandi acuminis, ac in administrandis observationibus exercitatissimus.

Finding that the tables of Ptolomy were imperfect, he computed new ones, which were long used as the best among the Arabs: these were adapted to the meridian of Aracta or Racah. Albategni composed in Arabic a work under the title of The Science of the Stars, comprising all parts of astronomy, according to his own observations and those of Ptolomy. This work, translated into Latin by Plato of Tibur, was published at Nuremberg in 1537, with some additions and demonstrations of Regiomontanus; and the same was reprinted at Bologna in 1645, with this author's notes. Dr. Halley detected many faults in these editions: Philos. Trans. for 1693, N° 204.

In this work, Albategni gives the motion of the sun's apogee since Ptolomy's time, as well as the motion of the stars, which he makes 1 degree in 70 years. He made the longitude of the sirst star of Aries to be 18° 2′; and the obliquity of the ecliptic 23° 35′. And upon Albategni's observations were founded the Alphonsine tables of the moon's motions; as is observed by Nic. Muler, in the Tab. Frisicæ, pa. 248.

ALBERTUS Magnus, a very learned man in the 13th century, who, among a multitude of books, wrote several upon the various mathematical sciences, as Arithmetic, Geometry, Perspective or Optics, Music, Astrology and Astronomy, particularly under the titles, de sphæra, de astris, de astronomia, item speculum astronomicum.

Albertus Magnus was born at Lawingen on the Danube, in Suabia, in 1205, or according to some in 1193; and he died at a great age, at Cologn, November 15, 1280. Vossius and other authors speak of him as a great genius, and deeply skilled in all the learning of the age. His writings were so numerous, that they make 21 volumes in folio, in the Lyons edition of 1615. He has passed also for the author of some writings relating to midwifery, &c, under the title of De natura rerum, and De secretis mulierum, in which there are many phrases and expressions unavoidable on such a subject, which gave great offence, and raised a clamour against him as the supposed author, and inconsistent with his character, being a Dominican friar, and sometime bishop of Ratisbon; which dignity however he soon resigned, through his love for solitude, to enter again into the monastic life. But the advocates of Albert assert, that he was not the author of either of these two works. It must be acknowledged however, that there are, in his Comment upon the Master of Sentences, some questions concerning the practice of conjugal duty, in which he has used some words rather too gross for chaste and delicate ears: but they allege what he himself used to say in his own vindication, that he came to the knowledge of so many monstrous things at confession, that it was impossible to avoid touching upon such questions. Albert was certainly a man of a most curious and inquisitive turn of mind, which gave rise to other accusations against him; such as, that he laboured to find out the philosopher's stone; that he was a magician; and that he made a machine in the shape of a man, which was an oracle to him, and explained all the difficulties he proposed: the common cant accusations of those times of ignorance and superstition. But having great knowledge in the mathematics and mechanics, by his skill in these sciences he probably formed a head, with springs capable of articulate sounds; like the machines of Boetius and others.

John Matthæus de Luna, in his treatise De Rerum Inventor<*>bus, has attributed the invention of fire-arms to Albert; but in this he is refuted by Naude, in his Apologie des grands hommes.

, otherwise called Abuassar, and Japhar, was a celebrated Arabian philosopher and astrologer, of the 9th or 10th century, or according to some authors much earlier. Blancanus, Vossius, &c, speak of him as one of the most learned astronomers of his time, or astrologers, which was then the same thing. He wrote a work De Magnis Conjunctionibus Annorum Revolutionibus, ac eorum Perfectionibus, printed at Venice in 1515, at the expence of Melchior Sessa, a work chiefly astrological.

He wrote also Introductio in Astronomiam, printed in the year 1489. And it is reported that he observed a comet in his time, above the orb of Venus.

, in the Arabian Astrology, is when a heavy slow-moving planet receives another lighter one within its orb, so as to come in conjunction with it.

, an Arabian name of a fixed star, of the first magnitude, just in the eye of the sign or constellation Taurus, or the bull, and hence it is popularly called the bull's eye. For the beginning of the year 1800, its Right Ascension is—66°6′51″.10Annual variation in AR 0051.31Declination—16552.00 N.And Annual variat. in Decl. 00 8.30

, a star of the third magnitude in the right shoulder of the constellation Cepheus.

, or Aldhaphra, in the Arabian Astronomy, denotes a fixed star of the third magnitude, in the mane of the sign or constellation Leo, the lion.

, an eminent French mathematician and philosopher, and one of the brightest ornaments of the 18th century. He was perpetual secretary to the French Academy of Sciences, and a member of most of the philosophical academies and societies of Europe.

D'Alembert was born at Paris, the 16th of November 1717. He derived the name of John le Rond from that of the church near which, after his birth, he was exposed as a foundling. But his father, informed of this circumstance, listening to the voice of nature and duty, took measures for the proper education of his child, and for his future subsistence in a state of ease and independence. His mother, it is said, was a lady of of rank, the celebrated Mademoiselle Tencin, sister to cardinal Tencin, archbishop of Lyons.

He received his first education among the Jansenists, in the College of the Four Nations, where he gave early signs of genius and capacity. In the first year of his philosophical studies, he composed a Commentary on the Epistle of St. Paul to the Romans. The Jansenists considered this production as an omen, that portended to the party of Port-Royal a restoration to some part of their former splendor, and hoped to find one day in d'Alembert a second Pascal. To render this resemblance more complete, they engaged their pupil in the study of the mathematics; but they soon perceived that his growing attachment to this science was likely to disappoint the hopes they had formed with respect to his future destination: they therefore endeavoured to divert him from this line; but their endeavours were fruitless.

On his quitting the college, finding himself alone, and unconnected in the world, he sought an asylum in the house of his nurse. He hoped that his fortune, though not ample, would enlarge the subsistence, and better the condition of her family, which was the only one that he could consider as his own. It was here therefore that he fixed his residence, resolving to apply himself entirely to the study of geometry.—And here he lived, during the space of 40 years, with the greatest fimplicity, discovering the augmentation of his means only by increasing displays of his beneficence, concealing his growing reputation and celebrity from these honest people, and making their plain and uncouth manners the subject of good-natured pleasantry and philosophical observation. His good nurse perceived his ardent activity; heard him mentioned as the writer of many books; but never took it into her head that he was a great man, and rather beheld him with a kind of compassion. “You will never, said she to him one day, be any thing but a philosopher—and what is a philosopher?—a fool, who toils and plagues himself all his life, that people may talk of him when he is dead.”

As d'Alembert's fortune did not far exceed the demands of necessity, his friends advised him to think of some profession that might enable him to increase it. He accordingly turned his views to the law, and took his degrees in that faculty: but soon after, abandoning this line, he applied himself to the study of medicine. Geometry however was always drawing him back to his former pursuits; so that after many ineffectual struggles to resist its attractions, he renounced all views of a lucrative profession, and gave himself up entirely to mathematics and poverty.

In the year 1741 he was admitted a member of the Academy of Sciences; for which distinguished literary promotion, at so early an age (24), he had prepared the way by correcting the errors of a celebrated work (The Analyse Demontrée of Reyneau), which was esteemed classical in France in the line of analytics. He afterwards set himself to examine, with close attention and assiduity, what must be the motion and path of a body, which passes from one fluid into another denser fluid, in a direction oblique to the surface between the two fluids. Every one knows the phenomenon which happens in this case, and amuses children, under the denomination of Ducks and Drakes; but it was d'Alembert who first explained it in a satisfactory and philosophical manner.

Two years after his election to a place in the academy, he published his Treatise on Dynamics. The new principle developed in this treatise, consisted in establishing an equality, at each instant, between the changes that the motion of a body has undergone, and the forces or powers which have been employed to produce them: or, to express the same thing otherwise, in separating into two parts the action of the moving powers, and considering the one as producing alone the motion of the body, in the second instant, and the other as employed to destroy that which it had in the first.

So early as the year 1744, d'Alembert had applied this principle to the theory of the equilibrium, and the motion of fluids: and all the problems before resolved in physics, became in some measure its corollaries. The discovery of this new principle was followed by that of a new calculus, the first essays of which were published in a Discourse on the General Theory of the Winds, to which the prize-medal was adjudged by the Academy of Berlin in the year 1746, which proved a new and brilliant addition to the fame of d'Alembert. This new calculus of Partial Differences he applied, the year following, to the problem of vibrating chords, the resolution of which, as well as the theory of the oscillations of the air and the propagation of sound, had been but imperfectly given by the mathematicians who preceded him; and these were his masters or his rivals.

In the year 1749 he furnished a method of applying his principle to the motion of any body of a given figure. He also resolved the problem of the precession of the equinoxes; determining its quantity, and explaining the phenomenon of the nutation of the terrestrial axis discovered by Dr. Bradley.

In 1752, d'Alembert published a treatise on the Resistance of Fluids. to which he gave the modest title of an Essay; though it contains a multitude of original ideas and new observations. About the same time he published, in the Memoirs of the Academy of Berlin, Researches concerning the Integral Calculus, which is greatly indebted to him for the rapid progress it has made in the present century.

While the studies of d'Alembert were confined to mere mathematics, he was little known or celebrated in his native country. His connections were limited to a small society of select friends. But his cheerful conversation, his smart and lively sallies, a happy knack at telling a story, a singular mixture of malice of speech with goodness of heart, and of delicacy of wit with simplicity of manners, rendering him a pleasing and interesting companion, his company began to be much sought after in the fashionable circles. His reputation at length made its way to the throne, and rendered him the object of royal attention and beneficence. The consequence was a pension from government, which he owed to the friendship of count d'Argenson.

But the tranquillity of d'Alembert was abated when his same grew more extensive, and when it was known beyond the circle of his friends, that a fine and enlightened taste for literature and philosophy accompanied his mathematical genius. Our author's eulogist ascribes to envy, detraction, &c, all the opposition and censure that d'Alembert met with on account of the famous Encyclopédie, or Dictionary of Arts and Sciences, in conjunction with Diderot. None surely will refuse the well-deserved tribute of applause to the eminent displays of genius, judgment, and true literary taste, with which d'Alembert has enriched that great work. Among others, the Preliminary Discourse he has prefixed to it, concerning the rise, progress, connections, and affinities of all the branches of human knowledge, is perhaps one of the most capital productions the philosophy of the age can boast of.

Some time after this, d'Alembert published his Philosophical, Historical, and Philological Miscellanies. These were followed by the Memoirs of Christina queen of Sweden; in which d'Alembert shewed that he was acquainted with the natural rights of mankind, and was bold enough to assert them. His Essay on the Intercourse of Men of Letters with Persons high in Rank and Ofsice, wounded the former to the quick, as it exposed to the eyes of the public the ignominy of those servile chains, which they feared to shake off, or were proud to wear. A lady of the court hearing one day the author accused of having exaggerated the despotism of the great, and the submission they require, answered slyly, “If he had consulted me, I would have told him still more of the matter.”

D'Alembert gave elegant specimens of his literary abilities in his translations of some select pieces of Tacitus. But these occupations did not divert him from his mathematical studies: for about the same time he enriched the Encyclopédie with a multitude of excellent articles in that line, and composed his Researches on several Important Points of the System of the World, in which he carried to a higher degree of perfection the solution of the problem concerning the perturbations of the planets, that had several years before been presented to the Academy.

In 1759 he published his Elements of Philosophy: a work much extolled as remarkable for its precision and perspicuity.

The resentment that was kindled (and the disputes that followed it) by the article Geneva, inserted in the Encyclopédie, are well known. D'Alembert did not leave this sield of controversy with flying colours. Voltaire was an auxiliary in the contest: but as he had no reputation to lose, in point of candour and decency; and as he weakened the blows of his enemies, by throwing both them and the spectators into fits of laughter, the issue of the war gave him little uneasiness. It fell more heavily on d'Alembert; and exposed him, even at home, to much contradiction and opposition.

It was on this occasion that the late king of Prussia offered him an honourable asylum at his court, and the office of president of his academy: and the king was not offended at d'Alembert's refusal of these distinctions, but cultivated an intimate friendship with him during the rest of his life. He had refused, some time before this, a proposal made by the empress of Russia to entrust him with the education of the Grand Duke; —a proposal accompanied with all the flattering offers that could tempt a man, ambitious of titles, or desirous of making an ample fortune: but the objects of his ambition were tranquillity and study.

In the year 1765, he published his Dissertation on the Destruction of the Jesuits. This piece drew upon him a swarm of adversaries, who only confirmed the merit and credit of his work by their manner of attacking it.

Beside the works already mentioned, he published nine volumes of memoirs and treatises, under the title of Opuscules; in which he has resolved a multitude of problems relating to astronomy, mathematics, and natural philosophy; of which his panegyrist, Condorcet, gives a particular account, more especially of those which exhibit new subjects, or new methods of investigation.

He published also Elements of Music; and rendered, at length, the system of Rameau intelligible: but he did not think the mathematical theory of the sonorous body sufficient to account for the rules of that art.

In the year 1772 he was chosen secretary to the French Academy of Sciences. He formed, soon after this preferment, the design of writing the lives of all the deceased academicians, from 1700 to 1772; and in the space of three years he executed this design, by composing 70 eulogies.

D'Alembert died on the 29th of October 1783, being nearly 66 years of age. In his moral character there were many amiable lines of candour, modesty, disinterestedness, and beneficence; which are described, with a diffusive detail, in his eulogium, by Condorcet, in the Hist. de l'Acad. Royale des Sciences, 1783.

As it may be curious and useful to have in one view an entire list of d'Alembert's writings, I shall here insert a catalogue of them, from Rozier's Nouvelle Table des Articles contenus dans les volumes de l'Academie Royale des Sciences de Paris, &c, as follows:

Traité de Dynamique, in 4to, Paris, 1743. The 2d ed. in 1758.

Traité de l'Equilibre et du Mouvement des Fluides. Paris, 1744; and the 2d edition in 1770.

Reflexions sur la Cause Générale des Vents; which gained the prize at Berlin in 1746; and was printed at Paris in 1747, in 4to.

Recherches sur la Précession des Équinoxes, & sur la Nutation de l'Axe de la Terre dans le Système Newtonien. Paris, 1749, in 4to.

Essais d'une Nouvelle Théorie du Mouvement des Fluides. Paris, 1752, in 4to.

Recherches sur differens Points importans du Système du Monde. Paris, 1754 and 1756, 3 vol. in 4to.

Elemens de Philosophie, 1759.

Opuscules Mathématiques, ou Memoires sur différens Sujets de Géométrie, de Méchaniques, d'Optiques, d'Astronomie. Paris, 9 vol. in 4to; 1761 to 1773.

Elémens de Musique, théorique & pratique, suivant les Principes de M. Rameau, eclairés, développés, & simplifiés. 1 vol. in 8vo. à Lyon.

De la Destruction des Jesuites, 1765.

In the Memoirs of the Academy of Paris are the following pieces, by d'Alembert: viz,

Précis de Dynamique, 1743, Hist. 164.

Précis de l'Equilibre & de Mouvement des Fluides, 1744, Hist. 55.

Methode générale pour déterminer les Orbites & les Mouvements de toutes les Planètes, en ayant égard à leur action mutuelle, 1745, p. 365.

Précis des Réflexions sur la Cause Générale des Vents, 1750, Hist. 41.

Précis des Recherches sur la Précession des Équinoxes, et sur la Nutation de l'Axe de la Terre dans le Système Newtonien, 1750, Hist. 134.

Essai d'une Nouvelle Théorie sur la Résistance des Fluides, 1752, Hist. 116.

Précis des Essais d'une Nouvelle Théorie de la Résistance des Fluides, 1753, Hist. 289.

Précis des Recherches sur les differens Points importans du Système du Monde, 1754, Hist. 125.

Recherches sur la Précession des Equinoxes, & sur la Nutation de l'Axe de la Terre, dans l'Hypothese de la Dissimilitude des Méridiens, 1754, p. 413, Hist. 116.

Reponse à un Article du Mémoire de M. l'Abbé de la Caille, su<*> la Théorie du Soleil, 1757, p. 145, Hist. 118.

Addition à ce Mémoire, 1757, p. 567, Hist. 118.

Précis des Opuscules Mathématiques, 1761, Hist. 86.

Précis du troisième volume des Opuscules Mathématiques, 1764, Hist. 92.

Nouvelles Recherches sur les Verres Optiques, pour servir de suite à la théorie qui en à été donnée dans le volume 3e des Opuscules Mathématiques. Premier Mémoire, 1764, p. 75, Hist. 175.

Nouvelles Recherches sur les Verres Optiques, pour fervir de suite à la théorie qui en a été donnée dans le troisième volume des Opuscules Mathématiques. Second Mémoire, 1765, p. 53.

Observations sur les Lunettes Achromatiques, 1765, p. 53, Hist. 119.

Suite des Recherches sur les Verres Optiques. Troisième Mémoire, 1767, p. 43, Hist. 153.

Recherches sur le Calcul Intégral, 1767, p. 573.

Accident arrivé par l'Explosion d'une Meule d'Emouleur, 1768, Hist. 31.

Précis des Opuscules de Mathématiques, 4e & 5e volumes. Leur Analyse, 1768, Hist. 83.

Recherches sur les Mouvemens de l'Axe d'une Planete quelconque dans l'hypothese de la Dissimilitude des Méridienes, 1768 p. 1, Hist. 95.

Suite des Recherches sur les Mouvemens, &c, 1768 p. 332, Hist. 95.

Recherches sur le Calcul Intégral, 1769, p. 73.

Mémoire sur les Principes de la Mech. 1769, p. 278.

And in the Memoirs of the Academy of Berlin, are the following pieces, by our author: viz,

Recherches sur le Calcul Intégral, premiere partie, 1746.

Solution de quelques problemes d'astronomie, 1747.

Recherches sur la cour be que forme une Corde Tendue, mise en Vibration, 1747.

Suite des recherches sur le Calcul Intégral, 1748.

Lettre à M. de Maupertuis, 1749.

Addition aux recherches sur la courbe que forme une Corde Tendue mise en Vibration, 1750.

Addition aux recherches sur le Calcul Intégral, 1750.

Lettre à M. le professeur Formey, 1755.

Extr. de differ. lettres à M. de la Grange, 1763.

Sur les Tautochrones, 1765.

Extr. de differ. lettres à M. de la Grange, 1769.

Also in the Memoirs of Turin are,

Differentes Lettres à M. de la Grange, en 1764 & 1765, tom. 3 of these Memoris.

Recherches sur differens sujets de Math. t. 4.

, or Alfeta, a name given to the star commonly called Lucida Coronæ.

, Alfergani, or Fargani, a celebrated Arabic astronomer, who flourished about the year 800. He was so called from the place of his nativity, Fergan, in Sogdiana, now called Maracanda, or Samarcand, anciently a part of Bactria. He is also called Ahmed (or Muhammed) ben-Cothair, or Katir. He wrote the Elements of Astronomy, in 30 chapters or sections. In this work the author chiefly follows Ptolomy, using the same hypotheses, and the same terms, and frequently citing him.

There are three Latin translations of Alfragan's work. The first was made in the 12th century, by Joannes Hispalensis; and was published at Ferrara in 1493, and at Nuremberg in 1537, with a preface by Melancthon. The second was by James Christman, from the Hebrew version of James Antoli, and appeared at Frankfort in 1590. Christman added to the first chapter of the work an ample commentary, in which he compares together the calendars of the Romans, the Egyptians, the Arabians, the Persians, the Syrians, and the Hebrews, and shews the correspondence of their years.

The third and best translation was made by Golius, professor of mathematics and Oriental languages at Leyden: this work, which came out in 1669, after the death of Golius, is accompanied with the Arabic text, and many learned notes upon the first nine chapters; for this author was not spared to carry them farther.

, commonly called Count Algaroti, a celebrated Italian of the present century, well skilled in Architecture and the Newtonian philosophy, &c. Algaroti was born at Padua, but in what year has not been mentioned. Led by curiosity, as well as a desire of improvement, he travelled early into foreign countries; and was very young when he arrived in France in 1736. It was here that he composed his Newtonian Philosophy for the Ladies, as Fontenelle had done his Cartesian Astronomy, in the work intitled The Plurality of Worlds. He was much noticed by the king of Prussia, who conferred on him many marks of his esteem. He died at Pisa the 23d of May, 1764, and gave orders for his own mausoleum, with this inscription upon it; Hic jacet Algarotus, sed non omnis. He was esteemed to be well skilled in painting, sculpture, and architecture. His works, which are numerous, and upon a variety of subjects, abound with vivacity, elegance, and wit: a collection of them has lately been made, and printed at Leghorn; but that for which he is chiefly intitled to a place in this work is his Newtonian Philosophy for the Ladies, a sprightly, ingenious, and popular work.

, a general method of resolving mathematical problems by means of equations. Or, it is a method of performing the calculations of all sorts of quantities by means of general signs or characters. At first, numbers and things were expressed by their names at full length; but afterwards these were abridged, and the initials of the words used instead of them; and, as the art advanced farther, the letters of the alphabet came to be employed as general representations of all sorts of quantities; and other marks were gradually introduced, to express all sorts of operations and combinations; so as to entitle it to different appellations— universal arithmetic, and literal arithmetic, and the arithmetic of signs.

The etymology of the name, Algebra, is given in various ways. It is pretty certain, however, that the word is Arabian, and that from those people we had the name, as well as the art itself, as is testisied by Lucas le Burgo, the first European author whose treatise was printed on this art, and who also refers to former authors and masters, from whose writings he had learned it. The Arabic name he gives it, is Alghebra e Almucabala, which is explained to signify the art of restitution and comparison, or opposition and comparison, or resolution and equation, all which agree well enough with the nature of this art. Some however derive it from various other arabic words; as from Geber, a celebrated philosopher, chemist, and mathematician, to whom also they ascribe the invention of this science: some likewise derive it from the word Geber, which with the particle al, makes Algeber, which is purely Arabic, and signifies the reduction of broken numbers or fractions to integers.

But Peter Ramus, in the beginning of his Algebra, says “the name Algebra is Syriac, signifying the art and doctrine of an excellent man. For Geber, in Syriac, is a name applied to men, and is sometimes a term of honour, as master or doctor among us. That there was a certain learned mathematician, who sent his Algebra, written in the Syriac language, to Alexander the Great, and he named it Almucabala, that is, the book of dark or mysterious things, which others would rather call the doctrine of Algebra. And to this day the same book is in great estimation among the learned in the oriental nations, and by the Indians who cultivate this art it is called Aljabra, and Alboret; though the name of the author himself is not known.” But Ramus gives no authority for this singular paragraph. It has however on various occasions been distinguished by other names. Lucas Paciolus, or de Burgo, in Italy, called it l'Arte Magiore: ditta dal vulgo la Regola de la Cosa over Alghebra e Almucabala; calling it l'Arte Magiore, or the greater art, to distinguish it from common arithmetic, which is called l'Arte Minore, or the lesser art. It seems too that it had been long and commonly known in his country by the name Regola de la Cosa, or Rule of the Thing; from whence came our rule of coss, cosic numbers, and such like terms. Some of his countrymen followed his denomination of the art; but other Italian and Latin writers called it Regula rei & census, the rule of the thing and the product, or the root and the square, as the unknown quantity in their equations commonly ascended no higher than the square or second power. From this Italian word census, pronounced chensus, came the barbarous word zenzus, used by the Germans and others, for quadratics; with the several zenzic or square roots. And hence [scruple], [dram],