from which it is deduced, is rational or irrational. The above is the most general form that can be given to continued fractions, but there are few cases in which it is necessary to have any others than those whose numerators are unity, and signs +, and therefore it will be sufficient to consider the latter form only. The series of fractions formed of the first term, the first two terms, the first three terms, &c. of any continued fraction, are called converging fractions. To reduce any proposed rational JFraction to a continued Fraction. Divide the denominator by the numerator, then the divisor by the remainder, and so on; and the successive quotients will be the several denominators of the continued fraction, 1 being the numerator. If the numerator of the proposed fraction be greater than its denominator, the continued fraction will be preceded by an integer. Exam. Reduce {}}} to a continu nominator for the first fraction. For the second fraction, multiply the terms of the first by the se. cond denominator, adding 1 to the product of the denominator. For every succeeding one multiply the terms of the last by the corres. ponding denominator, and add to the products the terms of the last except one. From this it follows that the dif. ference of any two contiguous fractions is 1 divided by the product of their denominators, and that the difference between the product of the numerator of each, by the other's denominator, is also 1. To extract the Square Root by continued Fractions. Since the quantity that we have to represent by a continued fraction is irrational, the fraction which expresses it must consist of an indefinite number of terms; but still it has this property, that after a certain period, the denominators of the several fractions recur again in the same order as at first ; and consequently, atter having arrived at that period, the series may be carried on to any length at pleasure. Let it be required to extract the square root of 19 by continued fractions; we first find the greatest integer contained in V 19, which is 4; therefore, By this latter formula the square root 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 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 diminulion of tenuperature, 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. Thus, glass vessels, suddenly cooled after having been formed, are so very brittle that they hardly bear to be touched with any hard body. The cause of this effect is thus explained: When glass in fusion is very suddenly cooled, its external parts become solid first, and determine the magnitude of the whole piece, while it still remains fluid within. The internal part, as it cools, is disposed to contract still further, but its contraction is prevented by the resistance 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, Conti NUED Fractions. Conver G 1 No Lines, those which tend to a common point. Conver G ING Rays, those which tend to a common focus. Converg in G Series, those series whose ternus continually diminish. CON V E R S E, in Mathematics, commonly signities the same thing as reverse. Thus, one proposiuion is called the converse of another, when, after a conclusion is drawn from something supposed in the converse proposition, unat conclusion is supposed ; and then, that which in the other was supposed, is mow drawn as a conclusion from it. Converse propositions are not mecessarily true, but require a demonstration; and Euclid always demonstrates such as he has occasion for. An instance will shew this. 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 right-lined figures are equal, then if they be laid the one on the other, their boundary lines exactly coincide and agree. It is manifest 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 inipossible 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 ade of a globular body. 123 w Convex Lens Mirror. See Lexa Mirror, &c. CONVEXITY, the exterior or outward surface of a convex or round body. CONVEXO-Concave Lens, is one that is convex on one slue, and concave on the other. Convexo - Conver Lens, is one that is convex on both sides. CO-ORDINATES, in the theory of Curves, signify any absciss, and its corresponding ordinate. See ABscuss, OR DINATE, and CU R v M. COPERNICAN System, is that system of the world in which the sum is supposed at rest; and the earth and the several planets to revolve about him as a cenue, while the moon and the other satellites 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. C OR O L L A R Y, a consequent truth, which follows immediately from some pieceding truth or demonstration. CORPUSCLE, the diminutive of corpus, is used to denote the mimute particles that constitute natural bodies. CO R P US CULAR Philosophy, that schene 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. CORRECTION of a Fluent, in the Inverse Method of Fluations, is the determination of the constant or invariable quantity which belongs to the fluent of a given fluxion; 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 fluent of a may be either r, or r + c > and this, as well as the preelse value of the quantity e, must be determined from the nature of the oil. which may be done by the following rule. First, take" he fluent according to the proper rules for that pur. pose, and then observe whether this fluent becomes equal to zero, or to some constant quantity,when the nature of the problem requires that it should ; if it do, it is the complete 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 of c, which will be the correction 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 ar, it having been acted upon by some law Swhich rendered it subject to the fluxional calculus; and suppose the resulting fluxional equation to be w = a r + 3 are r the fluent of this is w = } are + aro + e c being the correction. Now, from the nature of the equation, when r = 0, or before the body had passed over any space, the velocity was a : therefore, when a = 'o, v = a , and writing these values we have a = 0 + c, or c = a, therefore the correct fluent is w = { a } + a a 3 + a Whereas, if the body had in the first instance been only solicited by gravity, then when a = 0, v would also have been equal to o ; and we should have found c = 0, or the general fluent would have been the correct fluent required. COSECANT, Cosine, Cotangent, cover; Sine, are the secant, sine, 26 ing an integer number. With ra- 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 crepusculum 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 l sphere. The crepuscula are occasioned by the sun’s rays refracted in our atmosphere, and reflected from the particles thereof to the eye. See Twl LIGHT. CROSS, an instrument used in surveying for the purpose of raising perpendiculars. It consists merely of two pair of sights set at o angles to each other, mounted on a staff, of a convenient height for use. CUBES, or Cube Numbers, in Arithmetic, and the Theory of JNumbers, 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. 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 + 1. Cube numbers divided by 6 leave the same remainder as their root, when divided by 6. quently the difference between any integral cube and its root is divisible by 6. Neither the sum nor difference of two cubes can be a cube, that is, the equation r" + y& = z* is impossible. The sum of a number of consecu. tive cubes beginning with unity in a square whose root is equal to the sum of the roots of all the cubes. The third differences of consecutive cubes are equal to each other, i.eing 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. Divide this dividend by 3 And conse 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 following 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. Ev. Required the cube root of 41278-212816 41218-242816(34:56 root 33 – 27 32 × 3–27) 14278(42d figure of the 41278 [root, 1st two pe. 343 = 39.304 [riods 342x3=3465) ign F212(5, 3d figure. 41278-242 1st three pe and thus the operation may be carried on till the root be obtained to any degree of accuracy required. This method, however, is extremely laborious, and is seldom or never employed; as other methods have been found in which the approximation is carried on much more rapidly. 2. Method. Write the given number under the form as + b, then by the binomial theorem, va'it) = a +*-** |