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sible combinations of every two sort, n of another sort, p of ana of them; the fourth of every ther sort, &c., by taking I al a three ; the fifth on every four, and time, 2 at a time, 3 at a time, &c. so on; that is, the number of com. to any given number of things at binations that may be formed on a time. of any number of things (n,) taken Rule. Place in ore horizontal a certain nuniber (m,) at a time, row m + 1 units, annexing ciphers will be expressed by the coefici. on the right hand, liil the whole ent of the m + 1 term of the above number of units and ciphers ex. expanded funciion, carried lo n ceeds the greatest number of lhings 1actors; or making a, b, c, d, &c. to be taken at a time by unity each equal to 1, the same will be Under each of these termos write expressed by the m + 1 coefficient the sum of n + 1, lett-hand terms, orihe expanded binomial (x +11;) including that as one of them un. which from the known law of the der which the number is placed ; binomial theorem is equal to

and under each of these, ihe p + 92 (1-1) (n—2)(n-3) · ·(n-m-1) : left-hand terms of the last line;

then the q + 1 terms of this, and 1.2.3.4.. m which is, therefore, a general for different things, and the last line

so on through all the number of mulæ for expressing the number will be the aviswer. of combinations that may be formed out of n things taken any num- the greatest use in the doctrine of

The theory of combinations is of ber m at a time.

chances and probabilities, for the Suppose, for example,

probability of an event happening required to find liow many combi. nations may be formed out of 13the number of combinations that

or failing depends generally upon cards, all different from eachiother, may be formed, or that may take takeni 4 at a time?

place amongst the circumstances 13. 12.11.10 Here we ve

on whicli the event ultimately de. 1.2.3.4

pends. 715 combinations.

COMBUSTIBLE Bodies, are The above formulæ leads us also those bodies which when once set to the method of finding the whole on lire will continue to burn withpossible number of combinations on the farther accession of fuel. that may be formed out of a given COMET, in Astronomy, a hea. number of things n, by taking 1,2, venly body, appearing at uncertain 3,4, &c. at a tinie, io n, at a time; periods, and which, during the for this it is obvious will be equal time of its appearance, has a mo. to the sum of all the coetlicients tion in some respects similar to of the expanded binomial (.+ 1^,! those of the planets. The orbits wanting the first terni. But the of the comels however differ from sum of all these coefficients is those of the planets, in their being equal to 2n, therefore the number more eccentric, and being inclined of possible combinations of n things to the plane of the ecliptics in anis equal to 2n-1. The single quan- gles of various magnitudes, the lities a, b, c, &c. being classed un- plane of some of them being nearly der the general term combination, coincident, and others nearly perfor the sake of analogy, if these pendicular to the plane of the ecbe excluded the formulæ will be liptic. The motions of comets are 21-(n + 1.)

also some of them direct, and Thus the number of all the com others retrograde, whereas those binations that may be formed out of the planets are all direct. of the five letters a, b, c, d, e= Comets are popularly divided 23-1= 31, or excluding the single into three distinct classes, vis. termis, the number of combinations beurded, tailed, and huiry comets ; is 25–6 = 26.

though this distinction relates To determine the number of rather to the circumstance under combinations that may be formed which they are seen, thai to any out of a given number of things, difference of the bodies theme in which there are in things of ouc I selves.

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Thus, when the comet is eastward upper extremity than near the of the sun, and moves from it, it centre of the comet. 6. Their tails is said to be bearded; when the are always transparent, and the comet is westward of the sun, and smallest stars are seen througlo sets after it, it is said to be tailed; them without any sensible dimiand wlien the sun and the comel nation of their light, and without are in opposition, the train is hid those effects of retraction which behind the body of the coinet, might be expected from viewing excepting a little which appears them through a visible medium, round it in the form of a border of which circumstance seems to iue hair, hence it is called hriry, and dicate that the lails are composed hence the name of comet is de- of extremely rare and attenuuled rived.

matter; but with regard to their The estimates that have been formation we may be said to be given of the magnitudes of coniets, totally ignorant, notwithstanding are not sufficiently accurate to be the numerous hypotheses that depended upon ; for it does not have been advanced on this subappear that they distinguish beject. The tail of the comet of 1811, tween the nucleus and the sur-when at its maximum, subtended rounding atmosphere. Some co- an angle of at least 16 degrees, and mets, from the apparent magnitude was computed to be at least 23 and distance compared, have been million miles in length. judged to be much larger than the The comet of 1759 is known to moon, and even equal to some of perform its revolution in 70 years, the primary planets. The diame nearly, whence it appears that its ter of that of 1744, when at the mean distance is about 18 times distance of the sun from us, mea that of the earth', or a little less sured about 1!, and therefore its than the mean distance of Uranus; diameter must be about three times but, in consequence of the great ecthe diameter of the earth : at ano-centricity of its orbit, its aphelion ther time the diameter of its nu. point, or greatest distance from cleus was nearly equal to that of the sun, is nearly double that of the Jupiter. Hence it has been con- above planet. The perihelion disjectured, that some of the solar tance of this comei is about 6 of eclipses recorded in history, that the mean distance of the earth, cannot be verified by calculation which being taken from 36, the from tables of the sun and moon, mean transverse axis of its orbii, have been occasioned by the in- leaves 35.4 for its aplielion disterposition of comets between the tance, which is nearly double the sun and the earth.

greatest distance of Uranus, and The principal phenomena which about four times that of Saturn. have been observed with respect It is extremely difficult to deterto the tails of comels are :1. Their mine, from computation, the elliplails appear the largest and bright-lic orbit of a comet, lo any degree est immediately after the passage of accuracy: for when the orbit is through their perihelion, or near- very eccentric, a very small error est approach to the sun. 2. The in the observation will change the tail of a comet always declines compuied orbit into a parabola, from a just opposition to the 5111, or hyperbola. Now, from the towards those parts which the thickness and inequality of the body or nucleus pass over, in its atmosphere with which the comet progress through its orbit. 3. This is surrounded, it is impossible to declination is the smallest when determine, with any precision, the head or nucleus approaches when either the limb or centre of nearest the sun, and is still less near the comet pass the wire at the the neucleus of the comet, than lime of observation.

And this un towards the extremity of the tail. certainty in the observations will 1. The tails are somewhat brighter subject the computed orbit to a and more distinctly defined in their great error. The only safe way convex than in their concave part. to get the period of comets, is to 9. They are also broader at the compare the elements of all those

which have been computed, and, which it transfers its motion to where you find they agree very another body. well, you may conclude that they COMMUTA LION, in Astronomy. are elements of the same comet, it Angle of commutalion is the disbeing so extremely improbable tance between the sun's true place that the orbits of two different seen from the earth, and the place comels should have the same incli. of a planet reduced to the ecliptic; nation, the same perihelion dis- | and is, therefore, found by subtance, and the places of the peri. i tracting the same longitude from helion and node the same. Thus, the heliocentric longitude of the knowing the periodic time, we get planet. the major axis of the ellipse; and COMPASS, or Mariner's Com. the perihelion distance being pass, an instrument used at sea by known, the minor axis will be inariners, to direct and ascertain known. When the elements of the course of their ship. The inthe orbits agree, the comets may vention of this instrument is combe the same, although the periodic monly ascribed to Flavio Gioia, or time should vary a little; as that Flavio, of Malphi, about the year may arise froni the attraction of 1302. the bodies in our system, and which The common construction of the may also aller all the other ele. mariner's compass is extremely ments in a small degree.

simple. It consists of a circular COMETARIUM, a machine for brass box, which contains a paper conveying an idea of the revolucard; on which is drawn the 32 tion of a comet about the sun. Il points of the compass; and this is contrived in such a manner, as card is fixed on a magnetic needle, by elliptical wheels to show the which always turns to the north, unequal motion of a comet in every except a small deviation which is part of its orbit.

variable at different places, and COMMENSURABLE, among geo- at the same place at different metricians, au appellation given to times. such quantities as are measured Azimuth COMPASS. This differs by one and the same common from the common sea compass nieasure

only in this, that the circumferCOMMENSURABLE Numbers, whe-ence of the card or box is divided ther iutegers, surds, or fractions, into degrees; also to the box is are such as can be measured or fitted an index with two sights, divided by some other number which are upright pieces of brass without any remainder; such are placed diametrically opposite to 12, and 18, as being measured by each other, having a slit down the Band 3: also 2 V 2, and 3 2, bé- middle of them, through which ing measured by ✓ 2.

the sun or star is to be viewed at COMMENSURABLR in Pouer, is the time of observation. The use said of right lines, when their of this instrument is to take the squares are measured by one and bearing of any celestial object, the same space or superficies. when it is in or above the horizon,

COMMENSURABLE Surds, those in order to find from the magnetithat being reduced to their least cal azimuth, or amplitude, the vaterms, become true figurative quan- riation of the needle. tities of their kind; and are there These are the thirty-two princi. fore as a rational quantity to a ra- pal points of divisions drawn on tional one.

the compass card ; and are other. COMMON Measure or Divisor, wise called Rhumbs ; each of which in Arithmetic, is that number has a particular denomination exwhich will divide two other num. pressed by means of the initials of bers without leaving a remainder; ine four first points, North, East, and the greatest of such divisors South, West. is called the greatest common mea Each point contains 11° 15', and sure, or greatest common divisor. is divided into 1 points, containing

COMMUNICATION of Motion, | 2o 4851. that act of a moving body by COMPASSES, or Pair of Com

passes, a mathematical instrumenti Spring Compasses, or Dividers, for describing circles, measuring are made of hardened steel, with and dividing lines, or tigures, &c. an arched head, which by its

The common COMPASSES consists springs opens the legs; the openof two sharp-pomted branches or ing being directed by a circular legs, joined together at the top. screw fastened to one of the legs,

Triangular COMPASSES ; the con let through the other, and worked struction of which is like that of with a nut. the common compasses, with the Geometry of the COMPASSES, a addition of a third leg or point, species of geometry invented by which has a wolion every way. M. Mascheroni, ot Milan, by which Their use is to take three points at all the elementary problems of once, and so to form triangles, and plane geonielry are performed by Jay down three positions of a map the compasses only, without the to be copied at the same time. use of the ruler ; it is, however,

Beam COMPASSES consist of a more ingenious than profound, and long straight beam or bar, carry. may be considered rather as ing two brass cursors; one of subject of curiosity than of real these being fixed at one end, the utility. other sliding along the beam,with a COMPLEMENT of an Arch or screw to fasten it on occasionally. Angle, is what it wants of 90 de

Elliptic Compasses, commonly grees. called a trammel, consists of a Arithmetical CoxPLEMENT, of a cross with two dovetail grooves, logarithm, is what the logarithm at right angles, and a ruler with wants of 10.00000, &c.; and the two dovetail knobs and a tracing easiest way to find it is, beginning point. The two knobs are adjusted at the left hand, lo subtract every to the local distance of the ellipse, tigure from 9, and the last from 10. and the distance between the re COMPLEMENT, in Astronomy, demotest one and the tracing point notes the distance of a star from is made equal to the semi-trans- the zenith. verse (the distance of the nearer COMPLEMENT of Life, a term knob from the same point being much used in the doctrine of lifeequal to the semiconjugate). Then annuities, by De Moivre; to de. the cross being laid with its centre note the number of years which a over that of the ellipse, and the given life wants of 86, this being knobs let into the grooves, the the age which he considered as turning round of the ruler' will the ulinost probable extent of life. trace the ellipse.

COMPLEMENTS of a PuralleloGerman COMPASSES, have their gram, are the two smaller paralleJegs a little bent outwards to- lograms made by drawing two wards the top; so that when shut right lines through a point in the the points only meet.

diagonal, and parallel to the sides Hair Compasses have an adjust of the parallelogram. In every ing screw attached to one of the parallelogram these complements legs, by means of which measures are equal to each other. niay be taken to a very great de COMPOSITE Number, is that gree of accuracy.

which is produced by the multipli. Proportional COMPASSES, are cation of two or more numbers or those in which the joint lies, not factors, and is thus distinguished at the end of the legs, but be from a prime number, which caniween the points terminating each not be so produced. leg. These are either simple or COMPOSITION, is a species of compound. In the former sort the reasoning by which we proceed centre or place of the joint is fix. from things that are known and ed ; so that one pair of them serves given, step by step, till we arrive only for one proportion. In the at others, which were before ancompound ones the joint may be known. set at any distance, and conso COMPOSITION of Forces, in Mequently any proportion whatever chanics, is the method of finding easily obtained.

the quantity and direction of a

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single force, equivalent to two or to denote the curvilinear vacuity
more forces of which the quantity of hollow bodies.
and direction are given. It is CONCAVE Lenses, or Mirrors,
thus distinguished from Resolution have either one or both sides con.
of Forces, which is the method of cave. See LENS and MIRROR.
resolving a given force into two or CONCAVITY, from corcave, the
inore forces, the combined effect hollow or vacuity of bodies.
of which shall be equivalent to the CONCAVO)-Concave Lens, is that
single given force.

which is concave on both sides.
COMPOSITION of Proportion, is CONCAYO-Conver Lens, is that
when, of four proportionals, the which is concave on one side, and
sum of the 1st and 2d is to the 21, I convex on the other.
as the sum of the 3d and 4th is lo CONCENTRIC, baving a com-
the 4th.

mon centre, as concentric circles, COMPOSITION of Ratios, in Arith- ellipses, &c. metic and Algebra, is performed CONCHOID, the name of a curve by multiplying the quantities or invented by Nicomedes; and exponents of two or more ratios hence commonly called the Con. together, which product is then choid of Nicomedes, which said to be compounded of all the much used by the ancienis in the other ratios whose exponents were construction of solid problems. multiplied.

It is thus constructed.

From a COMPOUND Interest, is that point draw any number of lines, which arises by the continual ad. each cutting another line, then dition of the interest to the prin- from this line set off equal distancipal as it becomes due.

ces, either toward the point, or in COMPOUND Motion, is that which the opposite direction; when the arises from the effect of several lines and the curve drawn through conspiring forces, which may these points is the conchoid. render it either rectilinear or cur There are two kinds of Cont hoid vilinear, according lo the nature the first, or external one, on the of the forces and the circumstan- side opposite to the point; and the ces under which they act.

second, or internal one, on the COMPOUND Quantities, in Algebra, side next the point of the se. are those connected together by cond there are two forms, 1st. the signs + and ; they are dis. pointed, when the curve passes betinguished into binomials, trino-iween the line and the point; and mials, &c. according to the iřum. 2d looped, when it passes beyond ber of terms of which they are the point. composed.

If a be the portion cut off from COMPOUND Ratio, is that which the lines, b the perpendicular disarises from the composition of ra- tance of the point from the line tios.

cut by the others, the angle COMPRESSIBILITY, that qua- which’any of the lines cut makes lity of a body by which it yields with the perpendicular, and Z the to the pressure of another body or total length of the line so cut; force, so as to be brought into 'a

6 narrower compass.

then Z -

a. In which The following table shows the

CUS. quantity of compression of these to answers for the first conchoid, fuids and mercury,

when the and for the second. thermometer was at 509 and baro The swell of architectural cometer 29 inches.

lumns is usually a tirst conchoid. Compress. of Mill. pts. Sp grav. Newton approves of the use of Spirits of wine . . 66

845 this curve for the trisection of an. Oil of olives

49
....918

gles, tinding two mean projrorRain water....

45

1090 tionals, and in the construction of Sea water

1028 problems, for which purposes it Mercury 3 . 13595

was employed by the ancients. CONČAVE, an expression used CONDENSATION, the act

1

4).

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