withstanding an exposure at intervals of many hours. BUSHEL, a measure for dry CALCULATION, a reckoning, the result of arithmetical operation; in allusion to the calculi, or little stones, used by the antients as we now use figures. CALCULUS, among mathematicians, denotes a certain way of performing mathematical opera tions, investigations, &c. Thus we say antecedental calculus, differential calculus, fluxional calculus, &c. CALCULUS of Derivations. See Derivations. CALCULUS Differentialis, Exponentialis, Integralis, of Partial Differences, of Variations, &c. See Differential, Exponential, and Integral. Fluxional CALCULUS. See Flux jons. Literal CALCULUS is the same as Algebra. Numeral CALCULUS is the same as Arithmetic. CALENDAR, CALENDARIUM, or KALENDAR, a distribution of time accommodated to the uses of life, or a table or almanac containing the order of the days, weeks, months, feasts, &c. happening throughout the year. The word is derived from calenda, anciently written in large characters at the head of each month. The days in the calendar were originally divided into octades, or periods of eight days; but afterwards, in imitation of the Jews, into periods of seven, agreeably to the Mosaic law. The Roman CALENDAR was first formed by Romulus, for the use of his followers and people. The year was first supposed to consist of only 304 days, which was divided into 10 months, some of these months containing 20 days, others 35 days, and some more: it began with the first of March, and ended with December. Numa reformed this calendar, and added the months of January and February, making it to commence on the first of January, and to consist of 355 days. But as this was evidently deficien, of the true C. goods, which, by an act of parliament, is to contain, 2150-42 cubic inches. year, he ordered an intercalation of 45 days to be made every four years in this manner, viz. every two years an additional month of 22 days between February and March; and at the end of each two years more, another month of 23 days; the month thus interposed being called Macedonius, or the intercalary February. Julius Cæsar, with the aid of Sosigines, a celebrated astronomer of those times, farther reformed the Roman calendar, from whence arose the Julian calendar, and the Julian, or old style. Finding that the sun performed his annual course in 8651 days nearly, he di vided the year into 365 days, but every fourth year 366 days, adding a day that year before the 24th of February. This was farther reformed by order of Pope Gregory XIII.; whence arose the terms Gregorian calendar and style, or new style. Gregorian CALENDAR, which, by means of epacts rightly disposed through the several months, determines the new and full moons, with the time of Easter, and the moveable feasts depending upon it. Corrected CALENDAR, is that which, rejecting all the apparatus of golden numbers, epacts, and dominical letters, determines the equinox, and the paschal full moon, with the moveable feasts depending upon it, by computation from astronomical tables. This calendar was introduced in the year 1700. CALENDS, or KALENDS, in the Roman chronology, the first day of every month. CALIBER, or CALIPER, properly denotes the diameter of any round or cylindrical body. CALIBER, or CALIPER Compasses, or simply Calipers, a sort of compass made with bowed legs, for the purpose of taking the diameter of any round body, or of a cylin der, whether external or internal. CAMERA, in Optics, a name given to those machines or contrivances by which the images of ob jects are pourtrayed by reflection verted. Let AB (Plate II. fig. 2) upon a plane. There are two kinds, be an object opposite to C, a lens the Camera Lucida, and the Came-in one side of a dark box, the rays ra Obscura. from it will fall inverted and proCAMERA LUCIDA. The instrument duce on the opposite side the now known by this name, and image ba. In order to procure a which is the invention of Dr. Wol- direct image, the rays are made to laston, is one of the most ingeni-impinge upon a mirror, placed at ous applications of optical reflec- an angle of 45°, and be reflected tion. The principal part of it con- from that upon the receiving sursists of a glass prism, whose sec-face. In observatories the Cametion is similar to the trapezium ra Obscura is usually made with a A, B, C, E, (Plate II. fig. 1.) A is a concave mirror placed at 45°, and right angle, C is obtuse, and B and the image received upon a convex E acute. The prism, which may be table, which shews all the parts about inch in the side, and equally vivid. Beautiful instances long, is fixed on a stand, and has are to be seen at Greenwich, and over the angle E a plate of metal, at the Calton-hill, Edinburgh. with a sight-hole, half covered by the angle of the prism. The ray of light P a, coming from the point P, will enter the face AB of the prism at right angles, and conse quently undergo no refraction, but proceed in a straight line, and impinge on the face BC, at a. There it will, in consequence of the smaliness of the angle of incidence, be wholly reflected, and again impinge upon the face CE at b, and there be again reflected. The two angles of reflection being a CANCER, Tropic of, a small circle of the sphere parallel to the equinoctial, and passing through the beginning of the sign Cancer. CAPILLARY Tubes, pipes whose canals or bores are exceedingly narrow, being so called from their resemblance to a hair in size. One of the most singular phenomenor of these tubes is, that if you take several of them of dif ferent sizes, open at both ends, and immerse them a little way into water, or any other fluid, it will immediately rise in the tubes to a considerable height above the surface of that into which they are immersed. CARDINAL Points, in Geography and Navigation, the four principal points of the compass; viz. east, west, north, and south. right angle, it will pass at right angles through AE, to the eye above the sight at E, and thus the point P will, to that eye, appear to be at P in the line a y. The eye Another phenomenon of these will, at the same time, see that tubes is, that such of them as will line through the point of the sight naturally discharge water only by which is uncovered by the prism.drops, when electrified, yields it If now any object be substituted in a perpetual stream. for P, and a sheet of paper for ay, the outlines of the object will be seen, and can be delineated on the paper. The Camera Lucida is exceedingly portable, and as the image is not reflected or refracted by a curve surface, the whole of it is equally clear. It is, however, difficult to procure glass for the prism completely free of internal refractions. A convex lens is sometimes placed over the sight, for the purpose of magnifying the image. CARDIOIDE, the name of a curve so denominated by Castilliani, from its resemblance to a heart, sapsia; the construction of which is as follows: Through one extremity of the diameter of the circle, draw a number of lines cutting the circle. Upon these set off without the cirCAMERA OBSCURA. This instru-cle parts equal to the diameter; ment is formed by admitting the then the curve passing through all light into a dark box or chamber, the points is the Cardioide. through a convex lens, from a concave mirror, or even through a small hole, and the image when received without reflection is in CARTESIAN Philosophy. The first principle of the Cartesian philosophy is this, "I think, therefore I am:" this is the foundation of Des Cartes's metaphysics: that on which his physics is built is, That nothing exists but substances."" Substance he makes of two kinds; the one that thinks, the other is extended; so that actual thought and actual extension make the essence of substance. The essence of matter being thus fixed in extension, Des Cartes concludes that there is no vacuum, nor any possibility of it in nature, but that the universe is absolutely full: by this principle, mere space is quite excluded; for extension being implied in the idea of space, matter is so too. Motion is defined to be the translation of a body from the neighbourhood of others that are in contact with it, and considered as at rest, to the neighbourhood of other bodies: by which the distinction is destroyed between motion that is absolute or real, and that which is relative or apparent. He main tains that the same quantity of motion is always preserved in the universe, because God must be supposed to act in the most constant and immutable manner; and hence also he deduces his three laws of motion. He supposes that God created matter of an indefinite extension, which he separated into small square portions or masses, full of angles that he impressed two motions on this matter; the one, by which each part revolved about its own centre; and another, by which an assemblage, or system of them, turned round a common centre. From whence arose as many different vortices, or eddies, as there were different masses of matter, thus moving about com mon centres. The consequence of these motions in each vortex, according to Des Cartes, is as follows:-The parts of matter could not thus move and revolve amongst one another, without having their angles gradually broken; and this continual friction of parts and angles must produce three elements: the first of these an infinitely fine dust, formed of the angles broken off; the second, the spheres remaining after all the angular parts are thus removed; and those par ticles not yet rendered smooth and spherical, but still retaining some of their angles and hamous parts from the third element. Now the first or subtilest element, according to the laws of motion, must occupy the centre of each system, or vortex, by reason of the smallness of its parts and this is the matter which constitutes the sun, and the fixed stars above, and the fire below. The second element, made up of spheres, forms the atmosphere, and all the matter between the earth and the fixed stars, in such sort, that the largest spheres are always next the circumference of the vortex, and the smallest next its centre. The third element, formed of the irregular particles, is the matter that composes the earth, and all terrestrial bodies, together with comets, spots in the sun, &c. He accounts for the gravity of terrestrial bodies from the centrifugal force of the ether revolving round the earth and upon the same general principles he pretends to explain the phenomena of themagnet, and to account for all the other operations in nature. CATACAUSTIC Curves, in the higher Geometry and in Optics, are the species of caustic curves formed by reflection, which are gene. rated in the following manner: If there be an infinite number of rays proceeding from the radiating point, and reflected by any given curve, so that the angles of incidence be equal to the angles of reflection; then the curve to which the reflected rays are always tangents, is the catacaustic or caustic by reflection: a caustic curve is that formed by joining the points of concurrence of the several reflected rays. If the reflected ray be produced, so that it becomes equal to the incident ray, the curve formed will be the evolute of the caustic, be ginning at the point where the first ray enters; and the portion of the caustic intercepted, is the differ ence of the two incident rays added to the difference of the two reflected ones. When the generating curve is a geometrical one, the caustic will be so too, and will always be rectifi able. The caustic of the circle is a cycloid, or epicycloid, formed by the revolution of a circle upon a circle. If the inside of a smooth bason, containing in it any white liquor, as milk, be placed in the sun's rays, or in a strong candle-light, it will exhibit a very perfect catacaustic curve. CATENARY, CATENARIA, in the higher Geometry, a mechanical curve, into which a chain or rope forms itself, by its own weight, when hung freely between two points of suspension. The equation of this curve is (z+x)9 y=ax hyp. log. or CATOPTRICS, that branch of Optics which illustrates the laws and properties of light, reflected from mirrors or specula. See the articles Reflection, Mirror, Light, and Vision. centre. CATOPTRIC Telescope, the same as Reflecting TELESCOPE. CENTRAL, Something relating to CENTRAL Forces are those forces which tends directly to or from a certain point or centre; being called, in the former case, centripetal force, and in the latter, centrifugal. The doctrine of central forces depends on the first Newtonian law of motion; viz. "Every body perseveres in its state of rest, or uniform motion in a right line, until a change is effected in it by the agency of some external force." M. de Moivre, in his "Miscel. Analyt." p. 231, as well as in Phil. Trans. has treated on this subject, and to him we owe many elegant theorems relating to the doctrine of central forces. Varignon, Maclaurin, Simpson, Euler, Emerson, and de l'Hopital, &c. have treated on this subject; to the latter of whom we owe the following general and comprehensive proposition; viz. 4g2; v2=g: 28 2gr Consequently, when the contriequal, the velocity of the body is petal and centrifugal forces are equal to that which it would acradius. quire in falling through half the moving in a circle, is as the versed 2. The central force of a body, it is as the square of that are directsine of an indefinitely small arc, or ly, and as the diameter inversely. 3. If two bodies revolve uniformly in different circles, their central forces are in the duplicate ratio of their veiocities directly, and the diameters or radii of the circles inversely; V2 v2 V2 v2 that is, F: fd=R 4. And hence, if the radii or diameters be reciprocally in the duplicate ratio of the velocities, the central forces will be reciprocally in the duplicate ratio of the radii, or directly as the fourth power of the velocities; that is, if V2: v2r: R, then F: ƒ= r2; R2 V4: vt. 5. The central forces are as the diameters of the circles directly, and squares of the periodic times inversely. For if c be the circumference described in the time t, with the velocity v; then the space ctv, or If a body of any determinate weight move uniformly round a using this value of centre, with any given velocity, rule, it becomes C 2= ; hence, in the third dic times are in the subduplicate the velocity in the circle whose ratio of the diameters or radii of radius is R. And F=f, then circle. same Or since we have found v = 26000, and t≈ 5078, these formulæ become. 20000, 5078✔✔ R for the required ve for the periodic 7. If the velocities be recipro- the periodic time in the cally as the distances from the centre, the central forces will be reciprocally as the cubes of the same distances, or directly as the cubes of the velocities. That is, if Vr: R, then is F: f3 R3 =V3: v3. 8. If the velocities be recipro-locity. cally in the subduplicate ratio of the central distances, the squares of the times will be as the cubes of the distances; for if V2 : v2 = : R, then is T2: t2- R3: rs. 9. Wherefore, if the forces be reciprocally as the squares of the central distances, the squares of the periodic times will be as the cubes of the distances; or when F:f=r2: R2, then is T2: 12 = RS: 3. From the preceding theorems, we may deduce the velocity and periodic time of a body revolving in a circle, by means of its own gravity, at any given distance from the earth's centre. Let g be the space through which a heavy body falls, at the surface of the earth, in the first second of time, or 10 feet; then 2 g will measure the force of gravity at the surface; and r being assumed for the earth's radius, the velocity of the body in a circle at its surface, in one second, will be 2600 feet nearly; the radius of the earth being taken 21000000 feet. Again, putting c = 3.14159, &c. we shall have time. In the case of the moon R=60r, we have therefore 3357 feet per second, 26000 days, nearly And in the same way the velocities of the planets, and their several periodic times may be determined, their distances being given; or their periodic times being given, their distances may be found by the converse operation; the periodic time of the earth's revolution, and its distance from the sun, being supposed known. though our first theorems related It may be proper to observe, that merely to circular motion, yet they are equally true for eliptic orbits. CENTRE, in a general sense, denotes a point equally remote from the extremes of a line, surface, or solid. CENTRE of Attraction of a Body, is that point into which, if all its matter was collected, its action upon any remote particle would still be the same. The common centre of attraction of two or more bodies, is sometimes |