N 19, and a 4, the calculation will stand thus:

By this latter formula the square, 100t of any irrational square may be easily extracted, particularly as the computation on the right hand is very readily supplied without setting down the steps of the operation, as is done in the above example, which is merely for the purpose of explanation. It is not, however, for the sake of the arithmetical extraction of the square root, that this method has been devised, but in consequence of its application to indeterminate equations of the second degree, which admit of no other general method of solution.

CONTINUED Proportion, is that which the consequent of the first ratio is the same as the antecedent of the second. Thus 2: 6 = 6 : 18 is a continued proportion.

1

&c. &c.

parts to approach nearer to each other, in which sense it stands opposed to dilatation or expansion.

Water and all aqueous fluids are gradually contracted by a diminu tion of temperature, until they arrive at a certain point, which is about 8° above the freezing point; but below that point they begin to expand, and continue to do so according as the temperature is lowered; and similar effects have been observed with regard to some metals. In speaking of contraction, a remarkable phenomenon, of considerable importance in manufactures, obtrudes itself on our notice; which is, the hardness that certain bodies acquire in consequence of a sudden contraction, and this is particularly the case with glass and some of the metals. CONTINUITY, Law of, is that Thus, glass vessels, suddenly coolby which variable quantities pas-ed after having been formed, are sing from one magnitude to ano- so very brittle that they hardly ther, pass through all the interme- bear to be touched with any hard diate magnitudes, without ever body. The cause of this effect is passing over any of them abruptly. thus explained: When glass in fuMany philosophers and metaphy- sion is very suddenly cooled, its sicians have asserted the probable external parts become solid first, conformity of natural operations and determine the magnitude of to this law; but father Boscovich the whole piece, while it still reproves that the law is universal. mains fluid within. The internal CONTRACTION, in Physics, the part, as it cools, is disposed to conthe extent or dimen-tract still further, but its contracv, or the causing its tion is prevented by the resistance

diminish

sions

of the external parts, which form an arch or vault round it, so that the whole is left in a state of constraint; and as soon as the equilibrium is disturbed in any one part, the whole aggregate is destroyed. CONVERGENT, or Converging, the tendency of different things, variously disposed, to one common point. It is also sometimes used to denote an approximation towards the real value of a thing. CONVERGING Fractions. See def. 2, CONTINUED Fractions. CONVERGING Lines, those which tend to a common point. CONVERGING Rays, those which

tend to a common focus.

CONVEX Lens Mirror. See LENS Mirror, &c.

CONVEXITY, the exterior or outward surface of a convex or round body.

CONVEXO-Concave Lens, is one that is convex on one side, and concave on the other.

CONVEXO Convex Lens, is one that is convex on both sides. CO-ORDINATES, in the theory of Curves, signify any absciss, and its corresponding ordinate. See ABSCISS, ORDINATE, and CURVE.

COPERNICAN System, is that system of the world in which the sun is supposed at rest; and the earth and the several planets to revolve about him as a cenue, while the moon and the other sateilites revolve about their respective primaries in the same manner.

The heavens and stars are here

supposed at rest; and that diurnal motion that they appear to have from east to west, is imputed to the earth's motion from W. to E.

COROLLAR Y, a consequent truth, which follows immediately from some pieceding truth or de

monstration.

CORPUSCLE, the diminutive of corpus, is used to denote the mimute particles that constitute natu

CONVERGING Series, those series whose terms continually diminish. CONVERS E, in Mathematics, commonly signines the same thing as reverse. Thus, one proposition is called the converse of another, when, after a conclusion is drawn from something supposed in the converse proposition, that conclusion is supposed; and then, that which in the other was supposed, is now drawn as a conclusion from it. Converse propositions are not necessarily true, but require a demonstration; and Euclid always de monstrates such as he has occasion for. An instance will shew this.ral bodies. If two right-lined figures are so exactly of a size and form (both respecting their sides and angles) that being laid one on the other, their boundary lines do exactly coincide and agree, then no one doubts that these figures are equal. Now try the converse. If two CORRECTION of a Fluent, in right-lined figures are equal, then the Inverse Method of Fluxions, is if they be laid the one on the the determination of the constant other, their boundary lines exact or invariable quantity which be ly coincide and agree. It is mani-longs to the fluent of a given fluxfest that this proposition, though the converse of the former, is by no means true. A triangle and a square may have equal areas; but it is impossible the sides of the former can all coincide with those of the latter

CONVERSION of Proportion, is when of four proportionals it is inferred that the 1st is to its excess above the 2d, as the 3d to its excess above the 4th.

CONVEX, round or curved, or protuberant outwards, as the out de of a globular body.

CORPUSCULAR Philosophy, that scheme or system of physics, in which the phenomena of bodies are accounted for, from the motion, rest, position, &c. of the corpuscles or atoms of which the bodies consist.

ion; and which does not naturally arise from that fluxion, but depends wholly on the nature of the problem whence the fluxion was deduced.

That such correction is frequently necessary will be obvious, if we consider, that the fluxion of æ is a, and the fluxion of x + c is also x, whatever be the value of the constant part e, and with whatever sign it is connected with the varia ble a. And hence, conversely, the

fluent of a may be either z, or

c; and this, as well as the preCise value of the quantity e, must be determined from the nature of the problem, which may be done by the following rule.

tangent, and versed sine of the complement of the arch or angle. Co being a contraction of the word complement.

If the factors of the binomial a

COTESIAN Theorem, in the higher Geometry, an appellation First, take he fluent according distinguishing an elegant property to the proper rules for that pur- of the circle discovered by Mr. pose, and then observe whether Cotes, and of great use in the inthis fluent becomes equal to zero, tegration of differentials by rationor to some constant quantity,when al fractions. The theorem is this: the nature of the problem requires that it should; if it do, it is then be required, the index n becomplete fluent, and no correction is necessary; but if not, it wants a correction, and this correction is the difference between the two general sides when reduced to that particular state. Hence, connect the constant, but indeterminate quantity c, with one side of the fluxional equation, found as above; then in this equation substitute for the variable quantities, such values as they are known to have at any particular state, place, or time, and then from that particular state of the equation find the value which will be the correction

of

C, required.

Suppose that a body is projected with a velocity of a feet per second from a given place, to determine its velocity, after it has passed over a certain space x, it haring been acted upon by some law which rendered it subject to the fluxional calculus; and suppose the resulting fluxional equation to be v = x x + 3 ax'x

the fluent of this is

v = x2 + ax3 + e c being the correction.

Now, from the nature of the equation, when x = 0, or before the body had passed over any space, the velocity was a; there fore, when x = 0, v = a; and writing these values we have ao+c. or c = a, therefore the correct fluent is

v = · } x22 + a x2 + a Whereas, if the body had in the first instance been only solicited by gravity, then when = o, v would also have been equal to o; and we should have found co, or the general fluent would have been the correct fluent required. COSECANT, Cosine, Cotangent, Coversed Sine, are the secant, sine,

ing an integer number. With radius = a describe a circle, and divide its circumference into as many equal parts as there are units in 2n, then in the radius, produced if necessary, take from the centre a distance = x, and from the extremity of this distance draw lines, and to all the points of divi sion in the circumference, these lines taken alternately shall be the factors sought.

COUNTERPOISE, any weight which, placed in opposition to another weight, produces an equili

brium.

COURSE, in Navigation, the point of the compass, or horizon, which the ship steers on, or the angle which the rhumb line on which it sails makes with the me ridian; and is sometimes reckoned in degrees, and sometimes in points and quarter points of the pass.

com

CRANE, in Mechanics, a machine for the raising of weights, the power of which depends upon a double cone, two barrels of dif ferent diameters, a combination of pullies or of wheel work, but more commonly of the last.

CREPUSCULUM, in Astronomy, twilight; the time from the first dawn or appearance of the morning to the rising sun; and again, between the setting of the sun and the last remains of day. The cre pusculum is usually computed to begin and end when the sun is about 18 degrees below the horizon; for then the stars of the sixth magnitude disappear in the morning, and appear in the evening. It is of longer duration in the solstices than in the equinoxes, and longer in an oblique than in a right

sphere. The crepuscula are occasioned by the sun's rays refracted in our atmosphere, and reflected from the particles thereof to the eye. See TWILIGHT.

CROSS, an instrument used in surveying for the purpose of raising perpendiculars. It consists merely of two pair of sights set at right angles to each other, mounted on a staff, of a convenient height for use.

CUBES, or Cube Numbers, in Arithmetic, and the Theory of Numbers, are those whose cube root is a complete integer.

All cube numbers are of one of the forms 4n, or 4n+ 1, that is, all cube numbers are either divisible by 4, or when divided by 4 leave 1 for the remainder.

times the square of the figure of the root above determined, and the first figure of the quotient will be the second figure of the root.

4. Subtract the cube of these two figures of the root from the first two periods on the left, and to the remainder annex the fol-, lowing period, for a new dividend, which divide as before; and so on till the whole is finished. And finally point off as many figures for integers as there are periods of integers in the proposed number. Ex. Required the cube root of 41278-242816 41278-242816(34.56 root

33 27

32 X 3=27) 14278(4 2d figure of the 41278 [root, Ist two pe39304

343[riod's 342x3=3468) 1974 242(5, 3d figure. 41278 242 1st three pe3453 41063.625 [riods 3452x3=357075) 214-617816(6 4th fi[gure

All cube numbers are of one of the forms 9n, or 9n+1; that is, they are either divisible by 9, or when divided by 9, they leave for a remainder i. Cube numbers divided by 6 leave and thus the operation may be the same remainder as their root, carried on till the root be obtainwhen divided by 6. And conse-ed to any degree of accuracy requently the difference between any quired. This method, however, is integral cube and its root is divi- extremely laborious, and is seldom sible by 6. thods have been found in which or never employed; as other methe approximation is carried on much more rapidly.

Neither the sum nor difference of two cubes can be a cube, that is, the equation x3 y3: z3 is impossible.

2 b!

The sum of a number of consecn number under the form a3±b, 2. Method. Write the given tive cubes beginning with unity in a square whose root is equal to the then by the binomial theorem, sum of the roots of all the cubes.√(a3b1) = a÷· The third differences of consecutive cubes are equal to each other, 5 h3 Leing each equal to 6.

To find the Cube Root of a given
Number.

1. Separate the given number into periods of three figures each, by putting a point over the place of units, another over the place of thousands, and so on over every third figure, to the left-hand in integer, and to the right in decimals; and then find the nearest cube root of the first period, and set it in the quotient.

2. Subtract the cube of the figure of the root, thus found, from the first period, to the lefthand, and annex the following period to the remainder for a divi dend.

3 a

3.6 ± + &c.;

22.5 8.64 3.6.9.12 all

in which formula substituting the values of a and b, the root may as above be found to any degree of accuracy required. It should be observed, however, that this formula can only be advantage. ously employed when the given number is only a little more or less than an exact cube.

Er. Extract the cube root of 1001. Here 1001 103+1, therefore a 10 and b = 1; consequently 31001 = 10+

5

&c.

3. Divide this dividend by 351-108

1

1 2.102

9.105

+

1001 10.003333-000001111 ways positive, being the square of +&c. 10.0033322.

4th Method, by logarithms. Divide the logarithm of the given number by 3, and the quotient will be the logarithm of the root; the natural number of which will

be the root itself.

Thus to find the cube root of 7867.

log. 7867= 3)3.897404
root 19715=

1.299135

b, but 27 as will be negative, or positive, according as a, in the proposed equation, is negative or positive, being the cube of a; in the latter case, each branch of the root has a real value, which may be ascertained by the proper rules; and in the former case, the same will have place, if as <16; but if a be negative, and as > be, then each branch taken sepa

CUBIC Equation, in Algebra, isrately becomes imaginary, and the that in which the highest power notwithstanding that the two to root cannot be found by this rule; of the unknown quantity rises together are equal to a real quanti the third degree, as 23 + ax2 + ty. This is what is termed the Having first reduced the equa-ons, the solution of which, by Irreducible Case in Cubic Equati tion to the form as ax = b Make xp+q, then

bx + c = 0.

axs

23p+3 pq (p + q) + gs a (p + q) Whence by addition, since p + 9=x, we have

x8 + ax = = p2 + (3pq + a) x qs

= b.

If, therefore, we assume a=— 3pq, we have p3 + q = c, that is, we obtain the two following equa

tions:

q2, = { b± √ (362 +27 a3) and, q =

in the same way,

p = √ { ¿b ± √ (362 + 1⁄2 a3.) }

Whence in the proposed equation, becomes

means of formula, has bid defiance to the attempts of many of the ablest mathematicians of modern times.

CULMINATION, in Astronomy, over the meridian, or its greatest the passage of any heavenly body altitude during its diurnal revolution.

bending or flexure; by which it CURVATURE of a Line, is its becomes a curve of any particular form and properties. Thus the nature of the curvature of the circle is such, that every point in the periphery is equally distant from a point within called the centre; and so the curvature of the same circle is everywhere the same, but the curvature of all other curves is continually varying.

The curvature of a circle is so much the more, as its radius is less, being always reciprocally as the radius and the curvature of other curves, is measured by the reciprocal of the radius of a circle, having the same degree of curvature as any curve has, at some certain point.

Every curve is bent from its sure of which is the same as that tangent by its curvature, the meaof the angle of contact formed by + = ✔ (3b × √ +6+3) × the curve and tangent. Now the

Which formula is commonly called Carden's Rule for Cubic Equations.

In the above formulab is al

same tangent is common to an infinite number of circles, or other curves, all touching it, and each other in the same point of contact. So that any curve may be touched by an infinite number of different