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fects, viz. enlargement of the bulk of bodies, the separation of their parts, &c. are such as must be produced by the introduction of something real ; and the abstraction of this principle may naturally produce the effects of cold, such as contraction of the bulk of bodies, agglutination, &c.; whereas it would be unnatural to suppose that a body contracts its bulk, as its parts come into closer contact, because something else has been introduced amongst them. With respect to the last supposition, viz. whether the effects of heat and those of cold be not owing to two distinct principles, a few arguments, and the equivocal result of a few experiments, have, at times, been adduced in support of it. But the general and prevailing opinion annong philosophers is, that a single element, called caloric, produces heat, or the effects of expanding bodies separating their parts, &c.; and that cold is only a relative expression; that is, meaning only the decrement of heat; so that real or absolute cold consists only in the total abstraction of caloric ; and, that such a point, viz. the zero of heat may be determined, has been shown by the experiments, the discoveries, and the calculations of some late eminent philosophers, viz. Irvine, Black, Crawford, and others. COLLIMATION, Line of, in a telescope, is a line passing through the intersection of those wires that are fixed in the focus, and the centre of the object glass. COLLISION, in Meckanics and Physics, is the meeting and mutual striking of two or more bodies, one of which, at least, is in motion. The most simple of the problems relating to collision, is that of a body proceeding to strike against another at rest, or moving before it with less velocity, or approaching towards it. Des Cartes supposing that the same quantity of absolute motion always exists in the world, concluded that the sum of the motions after the impact was equal to the sum of the motions before it. But the proposition is true only in the first and seco; of these cases; it is false 07
when the two bodies meet each other; for, in that case, the sum of the motions after the impact is equal to the difference of their motions before it, not to their Sunn.
If the impact of two perfectly hard bodies be direct, they wili, after impact, either remain at rest, or move on uniformly together with different velocities, according to the circumstances under which they met.
et B and b represent two per. fectly hard bodies, and let the velocity of B be represented by V, and that of b by v, which may be taken either positive or negative, according as b moves in the same direction as B, or contrary to that. direction, and it will be zero when b is at rest. This notation being understood, all the circumstance of the motion of the two bodies, after collision, will be expressed by the formula, velocity = BV -- bo, B + b
which being accommodated to the three circuinstances under which v may enter, becomes
These formula arise from the supposition of the bodies being perfectly hard, and consequently that the two after impact move on uniformly together as one mass. In cases of perfectly elastic bodies, other formulae have place which express the motion of each body separately, as in the following proposition.
If the impact of two perfectly elastic bodies be direct, their relative velocities will be the same both before and after impact, or they will recede from each other
B + b
when b was at rest before impact, that is, when v = 0.
1f a perfectly hard body impinge obliquely upon a perfectly hard and immoveable plane, it will, after collision, move along the plane.
And its velocity before impact
Is to its velocity after impact
Is to the cosine of the angle of impact. But if the body be elastic it will rebound from the plane with the same velocity, and at the zame angle with which it met it.
COLOUR, in natural philosophy, that property of bodies which affects the sight only ; or that property possessed by the elementary rays of light, separated by any means whatever, of exciting in us different sensations according to their different refrangibility. Thus colour may be considered in two respects, as it regards bodies in general, and as it is produced by solar light.
the velocity of B
the velocity of b
the velocity of B
the velocity of b
the velocity of b
Sir Isaac Newton's theory of light and colours is deduced from clear and decisive experiments. 1st. That lights which differ in colour, differ also in degrees of refrangibility. £d. That the light of the sun, not. withstanding its uniform appearance, consists of rays differently refrangible. 3d. That those rays which are more refrangible than others, are also more reflexible. 4th. That as the rays of light differ in degrees of refrangibility and reflexibility, so they also dis. fer in their disposition to exhibit this or that particular colour; and that colours are not qualifications of light derived from refractions or reflections of natural bodies, as was generally believed, but original and connate properties, which are different in different rays, some rays being disposed to exhibit a red colour and no other, and some a green and no other, and so of the rest of the prismatic colours. 5th. That the light of the sun consists of violet-making, indigomaking, blue-making, green-making, yellow-making, orange-making, and red-making rays; , and all of these are different in their degrees of refrangibility and reflexibility; for the rays which produce red colours are the least refrangible, and those that make the violet the most; and the rest are more or less refrangible as they approach either of these extremes, in the order already mentioned : that is, orange is least refrangible next to red, yellow next to orange, and so on ; so that to the same degree of refrangibility there ever belongs the same colour, and to the same colour the same degree of refrangibility. 6th. Every homogeneal ray, considered apart, is refracted according to one and the same rule, so that its sine of incidence is to its sine of refraction in a given ratio; that is, every different coloured ray has a different ratio belonging to it. 7th. The species of colour, and degree of refrangibility and re
flexibility, proper to any particular sort of rays, is not mutable by reflection or refraction from natural bodies, nor by any other cause that has been yet observed. When any one kind of rays has been separated from those of other kinds, it has obstinately retained its colours, notwithstanding all endeavours to bring about a change. 8th. Yet seeming transmutations of colours may be made, where there is any mixture of divers sorts of rays; for, in such mixtures, the component colours appear not, but, by their m, utually alloying each other, constitute an intermediate coiour. 9th. There are, therefore, two sorts of colour, the one original and simple, the other compounded of these ; and all the colours in the universe are either the colours of homogeneal, simple light, or compounded of these mixed together in certain proportions. The colours of simple light are, as we observed before, violet, indigo, blue, green, yellow, orange, and red, together with an indefinite variety of intermediate gradations. The colours of compounded light are disserently compounded of these simple rays, mixed in various proportions: thus a mixture of yellow-making and blue-making rays exhibits a green colour, and a mixture of red and yellow makes an orange ; and in any colour the same in specie with the primary ones may be produced by the composition of the two colours next adjacent in the series of colours generated by the prism, whereof the one is next more refrangible, and the other next less refrangible. But this is not the case with those which are situated at too great a distance; orange and indigo do not produce the internediate green, nor scarlet and green the in termediate yellow. 10th. The most surprising and wonderful composition of light is that of whiteness; there is no one sort of rays which can alone exhibit that colour: it is ever compounded, and to its composition all the aforesaid primary colours are requisite. , 11th. As whiteness is produced by a covious reflection of rays of 109 , -.
all sorts of colours, when there is a due proportion in the mixture; so, on the contrary, blackness is produced by a suffocation and abSorption of the incident light, which being stopped and suppresseq in the black body, is not refiected outward, but reflected and refracted within the body till it be stifled and lost.
COLURES, in Astronomy and Geography, two great circles supposed to intersect each other, at right angles, in the pole, and to pass through the solstitial and equinoctual points of the ecliptic ; that which passes through the former point being called solstitial colure, and the other the equinoctial colure.
COMBINATIONS, in Mathematics, denote the different collections that may be formed out of any given number of things, taken a certain number at a time, without regard to the order in which they may be arranged ; and are thus distinguished from permutations, or changes, which have reserence to the order in which the several quantities may be disposed.
In order to find the number of combinations that a given number of things will admit of, let us take the continued product of the factors, (r. -- a) (a + b) (a + c) (a + d,) &c. viz.
MATHEM ATICAL AND PHYSICAL SCIENCE.
sible combinations of every two of them : the fourth of every three ; the fifth ol every four, and so on ; that is, the mumber of combinations that may be formed out of any number of things (n,) taken a certain number (m,) at a time, will be expressed by the coeffici. ent of the m + 1 term of the above expanded function, carried to n factors; or making a, b, c, d, &c. each equal to 1, the same will be expressed by the m + 1 coefficient of the expanded binomial (a + 1" ;) which from the known law of the binomial theorem is equal to * (m—1) (n—2) (n-3) → • (n—m-1) 1. 2. 3.4 . . m. which is, therefore, a general formulae sor expressing the number of combinations that may be formed out of n things taken any number m at a time. Suppose, for example, it were required to find how many combi. nations may be formed out of 13 cards, all different from each other, taken 4 at a time? 13. 12. 11. 10
l. 2. 3.4
Here we have
715 combinations. The above formulae leads us also to the method of finding the whole possible number of combinations that may be formed out of a given number of things n, by taking 1,2, 3, 4, &c. at a time, to n, at a time; for this it is obvious will be equal to the sum of all the coefficients of the expanded binomial (a + 1n,) wanting the first term, But the sum of all these coefficients is equal to 2", therefore the number of possible combinations of n things is equal to 2n—1. The single quantities a, b, c, &c. being classed under the general term combination, for the sake of analogy, if these be excluded the formulae will be 2n—(n--|- 1.) Thus the number of all the com. binations that may be formed out of the five letters a, b, c, d, e = 2?–1 = 31, or excluding the single terms, the number of combinations is 25–6 = 26. To determine the number of combinations that may be formed out of a given number of things, in who've are in things of ouc - 10 - -
sort, n of another sort, p of another sort, &c., by taking 1 at a time, 2 at a time, 3 at a time, &c. to any given number of things at a time. Rule. Place in one horizontal row m + 1 units, annexing ciphers on the right hand, tiil the whole number of units and ciphers exceeds the greatest number of things to be taken at a time by unity Under each of these terms write the sum of n + 1, left-hand terms, including that as one of them under which the number is placed : and under each of these, the p + I left-hand terms of the last line; then the q + 1 terms of this, and so on through all the number of different things, and the last line will be the answer. The theory of combinations is of the greatest use in the dectrine of chances and probabilities, for the probability of an event happening or failing depends generally upon the number of combinations that may be formed, or that may take place amongst the circumstances on which the event ultimately depends. C O M BUST I B L E Bodies, are those bodies which when once set on lire will continue to burn without the farther accession of fuel. COMET, in Astronomy, a heavenly body, appearing at uncertain periods, and which, during the time of its appearance, has a mo. tion in some respects similar to those of the planets. The orbits of the comets however differ from those of the planets, in their being more eccentric, and being inclined to the plane of the ecliptics in angles of various magnitudes, the plane of some of them being nearly coincident, and others nearly perpendicular to the plane of the ecliptic. The motions of comets are also some of them direct, and others retrograde, whereas those of the planets are all direct. Comets are popularly divided into three distinct classes, viz. bearded, tailed, and hairy comets ; though this distinction relates rather to the circumstance under which they are seen, than to any difference of the bodies themselves.
Thus, when the comet is eastward of the sun, and moves from it, it is said to be bearded ; when the comet is westward of the sun, and sets after it, it is said to be tailed ; and when the sun and the comet are in opposition, the train is hid behind the body of the coinet, excepting a little which appears round it in the form of a border of hair, hence it is called hairy, and hence the name of comet is derived. The estimates that have been given of the magnitudes of comets, are not sufficiently accurate to be depended upon ; for it does not appear that they distinguish between the nucleus and the surrounding atmosphere. Some comets, from the apparent magnitude and distance compared, have been judged to be much larger than the moon, and even equal to some of the primary planets. The diameter of that of 1744, when at the distance of the sun from us, measured about 11, and therefore its dianmeter must be about three times the diameter of the earth : at another time the diameter of its nucleus was nearly equal to that of Jupiter. Hence it has been conjectured, that some of the solar eclipses recorded in history, that cannot be verified by calculation from tables of the sun and moon, have been occasioned by the interposition of comets between the sun and the earth. The principal phenomena which have been observed with respect to the tails of comets are : 1. Their tails appear the largest and brightest immediately after the passage through their perihelion, or nearest approach to the sun. 2. The tail of a comet always declines from a just opposition to the sun, towards those parts which the body or nucleus pass over, in its progress through its orbit. 3. This declination is the sumallest when the head or nucleus approaches nearest the sun, and is still less near the neucleus of the comet, than towards the extremity of the tail. 4. The tails are somewhat brighter and more distinctly defined in their convex than in their concave part. 5. They are also broader at the
upper extremity than near the centre of the comet. 6. Their tails are always transparent, and the smallest stars are seen through them without any sensible diminution of their light, and without those effects of refraction which might be expected from viewing them through a visible medium, which circumstance seems to indicate that the tails are composed of extremely rare and attenuated matter; but with regard to their formation we may be said to be totally ignorant, notwithstanding the numerous hypotheses that have been advanced on this subject. The tail of the comet of 18; 1, when at its maximum, subtended an angle of at least 16 degrees, and was computed to be at least 23 million miles in length. The comet of 1759 is known to perform its revolution in 70 years, nearly, whence it appears that its mean distance is about 18 times that of the earth', or a little less than the mean distance of Uranus; but, in consequence of the great eccentricity of its orbit, its aphelion point, or greatest distance from the sun, is nearly double that of the above planet. The perihelion distance of this comet is about 6 of the mean distance of the earth, which being taken from 36, the mean transverse axis of its orbit, leaves 35-4 for its aphelion dis
tance, which is nearly double the
greatest distance of Uranus, and about four times that of Saturn. It is extremely difficult to determille, from computation, the elliplic orbit of a comet, to any degree of accuracy: for when the orbit is very eccentric, a very small error in the observation will change the computed orbit into a parabola, or hyperbola. Now, from the thickness and inequality of the atmosphere with which the comet is surrounded, it is impossible to determine, with any precision, when either the limb or centre of the comet pass the wire at the time of observation. And this uncertainty in the observations will subject the computed orbit to a great error. The only safe way to get the period of comets, is to compare the elements of all those