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or not. But being elastic, it is necessarily affected by pressure, which reduces it into such a space, that the elasticity which re-acts against the compressing weight, is equal to that weight.

as the velocities with which equal balls are impelled, are directly as the square roots of the forces act ing upon them, we shall be able always to estimate the effect previous to any explosion taking The elasticity of the air exerts place: thus, if n be the number of itself equally in all ways; and times the air is condensed, then as when it is at liberty, and freed✔ 1000: ✔✅n from the cause which compresses ing from the it, it expands equally in all directions, in consequence of which it always assumes a spherical figure, in the interstices of the fluid in which it is lodged. This is evident in liquors placed under the receiver of an air-pump; for, 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 retain their spherical figure.

The expansion of air, by means of its elastic property, when only the compressing force is taken off or diminished, is found to be surprisingly great; it having been known, in certain experiments, to expand itself into 13,679 times its original space, and this solely by its own natural power, without the application of fire.

The elasticity of the air, under the same pressure, is increased by heat and diminished by cold, and that according to several very accurate experiments, at the rate of about one-435th part for every degree of heat of Fahrenheit's thermometer.

AIR-Gun, a pneumatic machine, which expels bullets or shot with great velocity and effect; its operation depends upon the elastic power of air, which we have seen above increases in proportion to the greater degree of condensation, Now the elastic power of fired gunpowder being equal to the pressure of 1000 atmospheres, or 1000 times greater than that of common air, it follows that in order to produce the same effect with an airgun as with a musket, the air must be compressed into onethousandth part of its natural bulk; and for all inferior degrees of condensation, the effect will be proportionally diminished; and

the velocity arisexplosion of gunpowder to that arising from condensation. If, therefore, the air be condensed twenty times, its velocity will be about one seventh of that arising from fired gunpow. der. In the air-gun, however, the reservoir of condensed air is commonly very large, in proportion to the tube which contains the ball, and its density will be very little altered by expanding through the narrow tube; and, consequently, the ball will be urged throughout by nearly the same uniform force with that of the first instant and hence the exploding power of condensed air is much more considerable than appears from the preceding estimation, being little less than that of gunpowder, even under a condensation of ten times, provided the reservoir be of any considerable magnitude.

AIR-Pipes, an invention intended for clearing the holds of ships, and other close places, of their foul air.

AIR-Pump,a pneumatic machine, which is of great use, in explaining and demonstrating the properties of air. It was invented by Otto de Guericke, consul of Magdeburgh, about the year 1654; and though it has since been much improved in form, its principle re. mains the same. The principal use of the air-pump is to extract the air from a vessel, which in that state is said to be exhausted, and the degree of exhaustion depends upon the goodness of the machine. The vesse!, which is called a receiver, is fitted to a plate on the air-pump, having a small orifice in the middle, which communicates by a tube with the barrels of the pump. These barrels are of metal, or of glass, in each of which works an air-tight piston, having a valve, which opens outwards when the piston is forced down towards the bottom of the barrel.

There is another valve to pre-strokes of the piston, after four, vent the air from being drawn back into the receiver. The elasti- strokes, and so on, according to city of the compressed air at the the powers of the ratio; that is, bottom of the barrel being greater such power of the ratio as is dethan that of the external atmo. noted by the number of the strokes. sphere, forces open the valve, and Universally, if s denote the sum escapes; and when the piston is of the contents of the receiver and again raised, air from the receiver barrel, and r that of the receiver escapes into the barrel. The two only without the barrel, and n any barrels act thus alternately, till number of strokes of the piston; the air in the receiver be so much then, the original density of the exhausted as not to have elasticity air being 1, the density after n sufficient for forcing open the strokes will be rn valves, and there the degree of exhaustion which the pump can produce terminates.

sn

that is, the nth

power of the ratio
From the same formulæ, namely
d the density, we easily

sary 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

(+)=d, by

taking the logarithm of this equation, it is n X log.· =log. of d;

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log. d log. d. rlog.r-log.s

log.s

=

In whatever manner or form this machine is made, its use and operation are much the same. The air in the receiver is diminished at each stroke of the piston, by derive a rule for finding the numthe quantity of the barrel or cyber of strokes of the piston, neceslinder full, and thus repeating the operation again and again, the air is rarefied to any proposed degree, a mercurial gage showing at any time what the degree of exhaustion is. But, supposing no vapour from moisture, &c. to rise in the receiver, the degree of exhaustion, after any number of strokes of the riston, may be determined by knowing the respective capacities and hence n=of the barrel and the receiver, including the tube of communication. For as every stroke diminishes the density in a constant proportion, namely, as much as the whole content exceeds that of the cylinder or barrel, the sum of as many diminutions as there are strokes of the piston, will show the whole diminution by all the strokes. Thus if the capacity of the barrel be equal to that of the receiver, then, the barrel being half the sum of the whole content, 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 first stroke, that is only of the original density: continuing this operation, it follows that the density of the remaining air will be after three

that is, divide the log. of the proposed density by the log. of the ra tio of the receiver to the sum of the receiver and barrel together, and the quotient will show the number of strokes of the piston requisite to produce the degree of exhaustion required.

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 piece of metal and a 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

squirting engine, the stream is discontinued between the several strokes.

ALGEBRA, a general method of solving problems and questions, whether arithmetical or mathematical, by means of symbols which have no fixed or determi nate values like the arithmetical figures, and which, by preserving their original form through all the steps of a calculation, however

duce, from the solutions of particular problems general formulæ for the solution of all problems of the same kind.

oysters live for 24 hours. That the AIR-Vessel, is a name given to heart of an eel taken out of the those metallic cylinders which body continues to beat for good are placed between the two forcpart of an hour, and that more ing-pumps in the improved firebriskly than in the air. That warm engines. The water is injected by blood, milk, gall, &c. undergo a the action of the pistons through considerable intumescence and two pipes, with valves, into this ebullition. That a mouse or other vessel; the air previously containanimal may be brought, by de- ed in it will be compressed by the grees, to survive longer in a rare- water, in proportion to the quantified air, than it does naturally. ty admitted, and by its spring force That air may retain its usual pres- the water into a pipe, which will sure, after it is become unfit for discharge a constant and equal respiration. That the eggs of silk-stream; whereas, in the common 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 gunpowder 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 co-long, enable the calculator to depiously, 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 The origin of algebra, like that bottom in a darkish body, leaving of other sciences of ancient date the upper part of the receiver and gradual progress, is not easily clear and transparent; and that on ascertainable. The earliest trea inclining the vessel sometimes to tise on that part of analytics, which one side, and sometimes to another, is properly called algebra, now the fume preserves its surface ho- extant, is that of Diophantus, a rizontal, after the nature of other Greek of Alexandria, who flourishfluids. That heat may be produced ed about the year 350, and who by attrition. That camphire will wrote thirteen books, though only not take fire; and that gunpowder, six "Arithmeticorum" of them though some of the grains of a are preserved. He is the only heap of it be kindled by a burning- Greek author on algebra whose glass, will not give fire to the con- works have been handed down to tiguous grains. That glow-worms us, though some traces of it aplose their light in proportion as the pear in the writings of some auair is exhausted, and, at length, thors much more ancient, as Arbecome totally obscure; but on re-chimedes, Euclid, Apollonius, &c.; admitting the air, they presently recover it again. That a bell, on being struck, is not heard to ring, or very faintly. That water freezes much more rapidly than in the open air, especially if the basin which contains it be floating in a chemical mixture, by which the vapour arising from the water can be absorbed. But that a syphon will not run. That electricity appears like the aurora borealis.

and we know that Hypatia wrote a commentary on the work of Diophantus. By what means the Arabs became possessed of this art is not known; but both the name and the science were transmitted to Europe, and particularly to Spain, by the Arabians or Saracens, about the year 1100, or probably a little earlier.

Italy, however, took the lead in the cultivation of the science,

After this, algebra became more generally known and improved, especially by many in Italy; and soon after, the first rule was there found out by Scipio Ferreus, for resolving one case of a compound cubic equation. But this science, as well as other branches of mathematics, was, in an eminent manner, cultivated and improved by Hieronymus Cardan of Bonomia, a very learned author, who, in the year 1545, published a book containing the whole doctrine of cubic equations, which had been in part revealed to him by Nicholas Tartalea.

after its introduction into Europe. that the even roots of them are Leonardo Bonacci, of Pisa, has impossible or nothing, as to comsome allusions to it in his arith-mon use. He was also acquainted metic written after his return from with the number and nature of Africa and the Levant, in 1202. the roots of an equation, and that From his manuscript, Lucus Pac-partly from the signs of the terms, cioli, or de Borgo, derived the ru-and partly from the magnitude and diments of that knowledge which relation of the co-efficients. He enabled him to compose his "Sum- knew that the number of positive ma Arithmeticæ et Geometria Pro- roots is equal to the number of portionumque et Proportionalita- changes of the signs of the terms. tum," which was published in That the co-efficient of the second Italian at Venice, in 1494, and terin of the equation, is the differ again in 1523. ence between the positive and negative roots: That when the second term is wanted, the sum of the negative roots is equal to the sum of the positive roots. How to compose equations that shall have given roots. That changing the signs of the even terms, changes the signs of all the roots. That the number of roots failed in pairs; or what we now call impossible roots, were always in pairs. change the equation from one form to another, by taking away any term out of it. To increase or di minish the roots of a given equa tion. It appears, also, that he had a rule for extraeting the cube root of such binomials as admit of ex traction; and that he often used the literal notation a, b, c, d, &c. That he gave a rule for biquadratic equations, suiting all their cases; and that in the investigation of that rule, he made use of an assumed indeterminate quantity; and afterwards found its value by the arbitrary assumption of a relation between the terms. Also, that he applied algebra to the resolution of geometrical problems; and was well acquainted with the difficulty of what is called the Irreducible Case, in attempting the solution of which he spent a good deal of time.

The chief improvements made by Cardan, as collected from his writings, are stated by Dr. Hutton as follows:

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Tartalea communicated to him only the rules for resolving three cases of cubic equations, namely; x2 + bx = c, x3 = bx + c, and as cbx; and he from thence raised a very large and complete work, laying down rules for all forms and varieties of cubic equations, having all their terms, or wanting any of them, and having all possible varieties of signs; demonstrating all these rules geometrically; and treating very fully of almost all sorts of transformations of equation, in a manner totally new. About the time that Cardan and It appears, also, that he was well Tartalea flourished in Italy, the acquainted with all the roots of science of algebra was cultivated equations, that are real, both posi- in Germany by Stifelius and tive and negative; or, as he calls Scheubelius. Stifelius introduced them, true and fictitious; and that the characters +, √, for plus, he made use of them both occa- minus, and root, or radix, as he sionally. He also shewed that the called it. Also the initials 2, 3, even roots of positive quantities are either positive or negative; V, for the powers 1, 2, 3, &c. He that the odd roots of negative quan- treated all the higher orders of tities are real and negative; but quadratics by the same general

He

rule; and introduced the numeral, introduced the general use of the exponents of the powers-3,-2,1, 0, 1, 2, 3, &c. both positive and negative, as far as integral numbers, but not fractional ones; calling them by the name exponens, exponent; and taught the general uses of the exponents in the several operations of powers, as we now use them in the logarithms. He likewise used the general literal notation A, B, C, D, &c. for so many different, unknown, or general quantities.

Scheubelius treats pretty largely upon surds, and gives a general rule for extracting the root of any binomial or residual, ab, where one or both parts are surds, and a the greater quantity; namely, that the square root of it is √a+√a-b

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which he illustrates by various examples.

A few years after the appearance of these treatises in Italy and Germany, Robert Recorde, a celebrated mathematician and physi cian, born in Wales, published "The Whetstone of Witte, which is the seconde part of Arithmetike: containing the Extraction of Rootes; the Cossike Practice, with the Rule of Equation; and the Works of Surde Nombers." He introduced the extraction of the roots of compound algebraic quantities; the use of the terms binomial and residual; the use of the sign of equality, or =.

Peletarius, in 1558, shewed that the root of an equation is one of the divisors of the last, or absolute term. He taught how to reduce trinomials to simple terms, by multiplying them by compound factors. He taught curious precepts and properties concerning square and cube numbers, and the method of constructing a series of each by addition only; namely, by adding successively their several orders of differences.

The science received further improvements up to the year 1600, from Ramus, Bombelli, Steven, and others; and a few years after, Schooten published the whole mathematical works of Vieta. Vieta

letters of the alphabet to denote
indefinite given quantities.
uses the vowels A, E, I, O, Y, for
the unknown quantities, and the
consonants, B, C, D, &c. for known
ones. He invented and introduced
many expressions or terms, several
of which are in use to this day:
such as co-efficient, affirmative and
negative, pure and adfected, or
affected, unciæ, homogeneum, &c.
and the line, or vinculum, over
compound quantities, thus, A+B.

Albert Girard was the first person who understood the general doctrine of the formation of the co-efficients of the powers, from. the sums of their roots, and their products, &c. He was the first who understood the use of nega tive roots, in the solution of geometrical problems; and was the first who spoke of the imaginary roots, and understood that every equation might have as many roots, real and imaginary, and no more, as there are units in the index of the highest power.

Thomas Harriot flourished about

the year 1610. He introduced the
uniform use of the small letters
a, b, c, d, &c.; viz. the vowels a,
e, and o, for unknown quantities,
and the consonants, b, c, d, f, &c.
for the known ones; which he
joins together like the letters of a
word, to represent the multiplica
tion or product of any number of
these literal quantities, and prefix.
ing the numeral co-efficient as we
do at present, except being sepa-
rated by a point, thus 5.bbc. For
a root he sets the index of the
root after the mark; as 3 for
the cube root. He also introduced
the characters> and <, for greater
and less; and in the reduction of
equations he arranged the opera
tions in separate steps, or lines,
setting the explanations in the
margin on the left-hand for line.
He may be considered as the
introducer of the modern state of
algebra. He also showed the uni-
versal generation of all the com.
pound or adfected equations, by
the continual multiplication of so
many binomial roots.

Oughtred's Clavis appeared in 1631, the same year in which Har

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