CALCAREOUS TUFA-CALC-SINTER. the primary form of its crystals being a rhomboid, of which the greatest angles are 105° 5'. Its secondary forms are more numerous than those of any Calcareous Spar.-Various Forms of Crystals. other mineral. More than seven hundred have been observed. One of the most common, a rather elongated pyramid, is sometimes called Dog-tooth Spar. C. S. is colourless and transparent, except in consequence of impurities which may be present in it; and when perfectly transparent, it exhibits in a high degree the property of double refraction of light, which was first discovered in it by Bartholinus. The presence of foreign substances frequently renders C. S. gray, blue, green, yellow, red, brown, or even black. The name Iceland Spar has often been given to C. S., at least to the finest colourless and transparent variety, because it is found in Iceland, massive in trap-rock. Slate Spar is a lamellar variety, often with a shining, pearly lustre, and a greasy feel, of which Wicklow in Ireland, and Glen Tilt in Scotland are localities. or CALCAREOUS TUFA, CALC-TUFF, TUFACEOUS LIMESTONE, a mineral which in its chemical composition is nearly identical with limestone and marble; but is distinguished by its spongy and cellular structure. It is generally rather soft, brittle, and friable, but sometimes it is sufficiently hard to be used as a building stone. The travertino, used for building at Rome, is a hard The colour of C. T. is generally yellowish-gray, sometimes yellow or yellowishIt occurs massive, or brown. uncrystalline forms, as tubular, botryoidal (like clusters of grapes), cellular, &c. Sometimes it incrusts It is frequent in animal and vegetable remains. the neighbourhood of calcareous springs. It is sometimes used as a filtering-stone. calcareous tufa. assumes many CALCEDONY. See CHALCEDONY. CALCEOLA'RIA (Lat. calceolus, a little shoe), a genus of plants of the natural order Scrophulariacea (q. v.), of which there are numerous species, natives of South America, chiefly of that part of the Andes which is more than 9600 feet above the sea, a few of them reaching almost to the utmost limits of vegetation; although some are found in lower and warmer situations, and some in the southern extremity of the American continent. They abound so much in some parts of Chili and Peru, as to give a peculiar aspect to the landscape. The calyx in this genus is 4-partite; the corolla, 2-lipped; the lower lip remarkably inflated, so as to form a bag; and the shape of the whole in some species considerably resembling that of a slipper. There are only two fertile stamens, and the capsule is semiSome of the species bivalvular with bifid valves. are shrubby, some herbaceous, almost all the herbaceous species being perennial. Many of them have corymbs of numerous showy flowers. Yellow is the colour which chiefly prevails in the flowers of the original species, and next to it purple; but the art of the gardener has succeeded in producing Calceolarias have been varieties and hybrids which exhibit many other rich and delicate tints. florists' flowers since about 1830, the curious appearance of the flowers combining with their beauty to render them attractive, and in no genus is the production of hybrids more easily or frequently effected. They are easily propagated by cuttings. Few plants require more liberal supplies of water. They are generally treated in Britain as half-hardy or as greenhouse plants. Some of the species are used in South America for dyeing. The roots of C. arachnoidea, a parent of many of the hybrids in our gardens, are largely employed in Chili, under the name of Relbum, for dyeing woollen cloths crimson. CALCINA'TION, or CALCI'NING (see CALX), is the process of heating or roasting in furnaces or It is resorted in heaps the various metallic ores. to as the first stage in the extraction of the majority of the common metals from their ores, and is essentially a process of oxidation. CA'LCIUM is the metal present in chalk, stucco, and other compounds of lime. It may be obtained by passing a powerful current of voltaic electricity through fused chloride of C. (CaCl), when the metal separates in minute globules. It is a yellowishwhite metal, can be rolled into sheets, and hammered into leaves, and is intermediate between lead and gold in hardness. It is represented by the symbol Ca, has the atomic weight or equivalent 20, and has the density 1.578, or nearly half as heavy again as water. At ordinary temperatures, it slowly tarnishes by oxidation; and when placed in contact with water, it rapidly decomposes the water (HO), forming lime (CaO), whilst hydrogen escapes. To be retained bright, C. must be kept under the surface of naphtha. At a red heat, it melts and burns with a dazzling white light, accompanied by scintillations. See LIME. CALCOTT, SIR AUGUSTUS WALL, R.A., a disKensington, London, in 1779. In 1803, he devoted tinguished English landscape painter, was born at himself to landscape painting; in 1810, was made a member of the Royal Academy; was knighted in 1837; and in 1844, made conservator of royal picHis landscapes are remarkable for their beauty, clear definition of objects, good drawing, and truthful natural colouring. He has been called not altogether unentitled. He died November 1844. the English Claude, a designation to which he is tures. CALCOTT, JOHN WALL, a distinguished musical Kensington, 1766. Too nervous to be a surgeon, for composer, elder brother of the above, was born at which he was intended, he devoted his attention to music, and in 1785 won three of the four gold medals annually given by the Catch Club, the admired O Sovereign of the willing Soul being one of the successful pieces. During the next ten years, In 1785, he was made Bachelor, and five he obtained twenty of the medals given by the same years afterwards, Doctor of music at Oxford. In society. 1805, he published his Musical Grammar; in the following year his mind gave way under the recovered again, but only for three years, when he continuous strain to which it had been subject. He relapsed, and continued insane until his death in May 1821. He was one of the most eminent composers belonging to the British school of music, and especially celebrated for his glee compositions. His choicest productions were published in two volumes by his son-in-law, Mr Horsley, in 1824. CA'LC-SINTER, a mineral, chemically identical with the purest marble and calcareous spar, but peculiarly characterised by its fibrous structure. It 503 CALCULATING MACHINE-CALCULUS. is formed from water holding carbonate of lime in solution, and occurs generally incrusting the roofs, walls, and floors of caves, particularly those in limestone rocks; often assuming curious and even fantastic forms. Macalister's Cave, in the Isle of Skye, and the limestone caves of Derbyshire, are the most celebrated British localities. But the stalactitic cave of Antiparos, in the Grecian Archipelago, is a far more famous locality for this mineral, which is often called Calcareous Alabaster, and used for the same purposes with the true alabaster (q. v.), to which it is in some respects preferable, particularly as not being liable to injury from exposure to the air. Volterra, in Tuscany, is another very famous locality for Calc-sinter. CALCULATING MACHINE. The most remarkable application hitherto made of machinery, is perhaps that through which it has been used to relieve the scientific inquirer to a very great extent of the fatigue of manipulating figures, which consumes so much of his time and energies. Various machines have been constructed for this purpose, differing in the extent of their faculties-to use words more suitable to thinking beings than to engines and somewhat in the principles of their construction. By the Arithmometer, for instance, a machine invented by M. Thomas of Colmar, all ordinary arithmetical operations are executed without fatigue to the operator; and by a machine contrived by M. M. Scheutz, which rests on the principle of Differences (q.v.), on the turning of a wheel, the successive terms of any series whose law may be confided to it, are produced-the machine at the same time printing a large proportion of its results, and thus providing for the accuracy of its tables. It is a fact of which the nation should be proud, that our countryman, Mr Babbage, is universally acknowledged as the instigating and guiding genius in the progress of these remarkable inventions. Among his inventions are a Difference Engine, of very comprehensive powers, indeed capable of managing series so complex that the differences of its terms do not reach zero until we ascend to the seventh order (vide art. DIFFERENCES, CALCULUS OF). An immense range of nautical and astronomical tables lie within the limits just defined; and the machine further tabulates approximately any series whatever that can be treated by the Method of Differences. While engaged in constructing the Difference Machine, Mr Babbage, probably through his increased experience of the capabilities of machinery, was led to form a new conception-that, namely, of the Analytical Machine. This has not yet been fully realised; but there is no doubt but that, with proper encouragement, Mr Babbage would successfully construct it. He has actually succeeded so far as to devise the means of making his machine perform all the elementary operations of addition, subtraction, multiplication, and division; and it is clear that all changes that can be produced on quantity are merely combinations of these. If, then, he could but make his machine perform these operations at command, and according to any special order, it could clearly develop any function whatever whose law is ascertained and fixed. A solution of this difficulty was suggested by the Jacquard Loom (q. v.), in which the cards oblige a machine capable of working any pattern to work out one particular pattern; and Mr Babbage having succeeded so far as to form a machine capable of executing any development, expects, by means of cards of operations, to compel his C. M. to work according to one fixed law, and no other. The withdrawal of the government aid, given to him for a series of years, has, however, much to the public regret and loss, prevented, let us hope only for a time, the realisation of his views. Both machines will be found described in the third volume of Taylor's Scientific Memoirs. [The anticipations above expressed have not been realised. Mr Babbage died in 1871, and nothing farther seems to have been done towards completing the Analytical Machine.] See CA'LCULUS, or STONE (in Medicine), a hard concretion formed within the animal body, in consequence of the deposition in the solid form of matters which usually remain in solution. CONCRETION. The concretions most commonly termed calculi are those formed in the kidneys or bladder (Urinary C.); and those formed in the gallbladder or biliary ducts (Biliary C.). Both of these give rise to very painful symptoms, and may even threaten life. Biliary C., or Gall-stone, may generally be presumed to exist when excessively severe pain suddenly arises in the right side beneath the border of the ribs, and when in a few hours jaundice comes on, shewing that some obstruction has existed to the outward flow of the bile. But the absolute proof that these symptoms depend on C. is often wanting. The pain is fortunately transitory, but is more severe while it lasts than almost any other known form of suffering, unless it be that of a C. in the kidney and ureter. It may be relieved by large doses of opium, but the remedy requires to be cautiously given, as even in medical hands fatal accidents have occurred. Gall-stones, when impacted in the ducts, sometimes have proved fatal; but much more frequently they find their way, sooner or later, into the intestines. They are almost invariably composed of cholesterine (q. v.), with colouring matter and mucus, arranged in layers in a semi-crystalline disposition. Urinary C. is a disease of all ages, but most common in advanced life and in the male sex. It is also very frequent in gouty persons, or among those who pursue sedentary occupations, and live freely. It is rare among those who live much in the open air, or who take much violent exercise, and use little animal food and wine. Among sailors, it is pecu. liarly rare. In certain parts of the country, the disease is said to be frequent, as in Norfolk, and perhaps along the east coast of Scotland. In India, too, where some of the predisposing circumstances mentioned above can hardly be said to prevail, stone is by no means uncommon. It would appear, therefore, that the predisposing causes of C. are still very imperfectly understood. In its early stages, the disease usually presents itself in the form of Gravel, shewn by the passage of numerous very small portions of gritty concretions, which may be observed in the urine as a deposit like sand, or like small grains of Cayenne pepper. When such deposits occur frequently, especially if they are present at the time of passing the urine, and not merely after it has cooled, there is reason to apprehend the formation of calculus. If, in these circumstances, there are pains of a dull character in the loins, with occasional twinges of sharper suffering, no time should be lost in seeking medical advice. If a fit of very severe pain should occur in a person for some time affected with gravel, if the urine be bloody, if agonising twinges, commencing in the loins, sting downwards into the thigh or the groin, it is probable that a stone has already formed in the kidney, and is being displaced towards the bladder. C. in the bladder is at first attended with little suffering, as compared with that caused by the stone in its passage downwards from the kidney; but unless removed or evacuated, the C. is sure to enlarge, and it then becomes the cause of one of the most painful diseases that afflict humanity. The existence of a stone in the bladder, CALCULUS. however, should never be taken for granted without The discovery of the tendency to urinary C. at an The chief varieties of urinary C. are-1. Uric acid (red sand); 2. Urates of ammonia, soda, lime, &c. (brick-dust sediment); 3. Phosphates of ammonia and magnesia, lime, &c.; 4. Oxalate of lime; 5. Carbonate of lime (chiefly in domestic animals); 6. Cystine; 7. Xanthic oxide (a very rare form, dis covered by Dr Marcet). Calculi are frequently found to be composed of numerous successive layers, having a perfectly distinct chemical composition. Urates and phosphates in particular frequently succeed each other, and form what is called an alternating calculus. Alternating Calculus-from Dr Marcet's Essay on a, uric acid nucleus; b, oxalate of lime; c, phosphates of lime, When C. has once fairly formed in the urinary CA'LCULUS, THE INFINITESIMAL, otherwise sometimes called the Transcendental Analysis, is a branch of mathematical science which commands, by one general method, the most difficult problems in geometry and physics. The merit of the invention of this powerful mathematical instrument has been claimed for Leibnitz, but is undoubtedly due with equal justice to Newton, who laid the foundations for it in that celebrated section of his Principia in which he demonstrates the chief theorems regarding the ultimate values or limits of the ratios of variable quantities. The view of one class of writers is, that these distinguished men invented the C. simultaneously and independently; and it is the fact that Leibnitz's system is unfolded from premises differing somewhat from those of Newton. See FLUXIONS. Another class of writers hold that Newton is the real inventor, and that to Leibnitz no more can be conceded than that he was the first who, using the suggestions of Newton's genius, gave a systematic statement to the principle of the transcendental analysis, and invented its appropriate symbolic language. He had the doctrine of limits before him when he wrote, and did little more than unfold more fully the logic of the processes therein The Infinitesimal C., both in its pure and applied suggested, and exhibit them in algebraical forms. forms, whether of geometry or mechanics, is a branch of the science of number; its symbols are of the same kind, are operated on according to the same laws, and lead to analogous results. It differs from the other branches of the science of number, such as arithmetic and algebra, in regarding number as continuous-i. e., as being capable of gradual growth and of infinitesimal increase, whereas they deal with finite and discontinuous numbers. It differs from ordinary algebra in another respect. In the latter, the values of unknown quantities, and their relations with each other, are detected by aid of equations established between these quantities directly; in the C., on the other hand, the equations between the quantities are not directly established, but are obtained by means of other equations primarily established, not between them, but certain derivatives from them, or elements of them. This artifice is most fertile, for it can be shewn that in the great majority of cases the relations of quantities concerned in any problem may more easily be inferred from equations between these their derivatives or elements than between themselves. It will be seen that the C. created a new notion of number-as continuous or growing. It is now necessary, in order to a proper conception of it, that a precise idea should be formed of a differential. The simplest idea of a differential is unquestionably that got by considering number as made up of infinitesimal elements, and a differential or 'infinitesimal' as being the value of the difference between a number at one stage of its growth and at another very near it. Every finite number being-in the posed of an infinite number of these infinitesimal view of the C. as first conceived by Leibnitz-comelements, certain axioms at once present themselves regarding infinitesimals; as, for instance, that a finite number of them has no value at all when added to a finite quantity.' Many other such axioms readily follow, from which, on this view, the whole theory of the infinitesimal C. may be constructed. But there are logical objections to this mode of forming the theory of the transcendental analysis, and of three views that have been propounded, that now universally accepted as the most logical, and as being capable of the easiest application, is that founded on the method of limits, already referred to as the invention of Newton. 505 The CALCULUS. meaning of a differential on this view will now be explained. dy dx Dy F(x + Dx) — F(x) F(x + Dx) − F(x) It is clear that the C. can be applied only where numbers may have the continuous character-i. e., It is clear that, in the general case, where they are or may be conceived as being variable. If two unknown quantities are connected at the limit will still be some function of x. Calling by a single equation only, we clearly have the condition satisfied, as where y and x are connected it F'(x), we have generally by the equation = (1) ข F(x), where F is a sign denoting some function of x, as tan. x, cos. x, x2, &c. This equality may be satisfied by innumerable values of y and x. One question which the C. solves is, how does y vary when ≈ varies? To solve it, and, at the same time, shew how the doctrine of limits affects the definition of a differential, suppose x, y, and x + Dx, y + Dy, to be two pairs of values of the variables which satisfy the above equation; then (2) y = = F(x), and (3) y + Dy = F(x + From (2) and (3) we have, by subtraction, (4) Dy = F(x + Dx) − F(x); whence we have the ratio = dy F(x). F(x) is dx called the first differential coefficient of y or F(x). Being a function of x, it may be again differentiated. The result is written day dx2 = F"(x), "(x) being the second differential coefficient of y or F(x); and again F(x) may be a function of x, and of the differential C. to shew how to obtain the so capable of differentiation. Now, it is the object various differentials of those few simple functions of Dx).quantity which are recognised in analysis, whether they are presented singly or in any form of combination. Such functions are the sum, difference, product, and quotient of variables, and their powers and roots; exponentials, logarithms; and direct and inverse circular functions. The C. so far is complete as we can differentiate any of those functions or any combination of them-whether the functions be explicit or implicit ; and with equal ease we may differentiate them a second or any number of times. This C. is capable of many interesting applications as to problems of maxima and minima, the tracing of curves, &c., which cannot here be particularly noticed. This ratio will generally change in value as Dx and Dy diminish, till, as they both vanish, which they must do simultaneously, it assumes the form Taking this form, it ceases to have a determinate actual value, and it is necessary to resort to the method of limits, to ascertain the value to which it was approaching, as De and Dy approached zero. Let, then, dx and dy be any quantities whose ratio is equal to the limiting ratio of the increments Dx, Dy, so that dy dx limit Dx as De and Dy approach zero. Then dx and dy are of the Integral C., instead of being general, are little better than artifices suited to particular cases; no popular view can be given of these. In many cases, integration is quite impossible. The explanation of integration by parts, by approximation, definite integrals, and singular solutions, is far beyond the scope of the present work. The reader is referred to any of the numerous text-books on the subject. The Integral C. has applications in almost every branch of mathematical and physical science. It is specially of use in determining the lengths of curved lines, the areas of curved surfaces, and the solid contents of regular solids of whatever form. The whole of the lunar and planetary theories may be described as an application of the integral C., especially of that branch of it which deals with the integration of differential equations. It is applied, too, in hydrostatics and hydrodynamics, and in the sciences of light, sound, and heat. In short, it is an instrument without which most of the leading triumphs in physical science could never have been achieved. CALCULUS OF VARIATIONS.-The foundation of this C. is a method of differentiation, but of quite a The idea of a differential being once compre-peculiar kind. As above explained, the object of hended, the reader will be able to understand, in a general way, the main divisions of the C., which we shall now briefly delineate. So much is clear from what has been stated, that there must be two main divisions-one by which, the primary quantities being known, we may determine their differentials; and another by which, knowing the differentials, we may detect the primary quantities. These divisions constitute the Differential C. and Integral C. respectively. 1. THE DIFFERENTIAL CALCULUS.-Recurring to the formula already given we know the differential C. is to determine the form which a function, such as F(x), will assume if x receive an indefinitely small increment, such as Dr. In the C. of variations, the object is to ascertain and lay down the laws of the changes supervening on a slight alteration of the form of the function, or should F(x) become Fx. This C. commands with ease a class of problems called problems of isoperi meters, which were formerly insoluble. It has also power over mechanical problems, and many depart ments of high physics cannot be touched without its aid. Mr Airy and Professor Jellet have both CALCULUS-CALCUTTA. written works on the subject, which may be consulted. CALCULUS OF FINITE DIFFERENCES, CALCULUS OF FUNCTIONS, and CALCULUS OF OPERATIONS.-For brief notices of these growths from the original Transcendental Analysis, see DIFFERENCES, FUNCTIONS, and OPERATIONS. CALCUTTA (Kali Ghatta, the ghaut or landingplace of the goddess Kali), the capital of the presidency of Bengal, and metropolis of British India, is situated on the left bank of the river Hooghly, an arm of the Ganges, in 22° 35′ N. lat., and 88° 27′ E. long., about 100 miles from the sea by the river. C. was founded by Governor Charnock in the year 1686, by the removal hither of the factories of the East India Company. In 1700, three villages surrounding the factories having been conferred upon the company by the emperor of Delhi, in recognition of a present made to Azim, a son of Aurungzebe, they were forthwith fortified, and received the name of Fort William, in honour of the reigning king; but the place was subsequently termed Calcutta, the name of one of the villages. In 1707, C. had acquired some importance as a town, and was made the seat of a presidency. In 1756, however, a great misfortune befell the rising town; it was unexpectedly attacked by Surajah Dowlah, the Nawaub of Bengal, and being abandoned by a number of those whose duty it was to defend the place, it was compelled to yield after undergoing a two days' siege. Only 146 men, however, fell into the enemy's hands; but these were treated with heartless cruelty. Cast at night into a confined cell, about 20 feet square-the notorious 'Black Hole' (q. v.)they endured the most unheard-of sufferings, and in the morning it was found that only 23 out of 146 had survived the horrors of that night. The city remained in the hands of the enemy until eight months afterwards, when Clive arrived in the country from England. In conjunction with Admiral Watson, Clive succeeded in recapturing the town, and afterwards concluded a peace with the Nawaub. Soon after this, and subsequent to the important victory of Plassey, the possessions of the East India Company were greatly extended by means of grants made by the emperor of Delhi, and C. once more resumed its career of progress, and advanced rapidly in prosperity. In 1852, C. was erected into a municipality, the proprietors paying assessments, and electing commissioners to apply the proceeds of these assessments in cleansing, improving, and embellishing the town. In 1837, the population of the town proper amounted to 229,700; in 1872, it had increased to 447,601, and if we include the suburban parts, the number will stand 892,429. Besides these, thousands of the three and a half millions who sleep at night in the surrounding districts of Hooghly and the twenty-four Pergunnahs, flock during the day to C., on foot, by boat, or by railway, to their daily toil. The inhabitants are mostly Hindus; but there is also a good proportion of Mohammedans. About 20,000 are Europeans; 20,000 Eurasians, or the progeny of white fathers with native mothers; and there is a considerable number of Armenians, Greeks, Jews, Parsees, and negroes. The city extends for about five miles along the river, and is somewhat less than two miles in breadth at its broadest part, the area being about eight square miles, and comprised for the most part between the river and the Circular Road, a spacious roadway which marks the landward Beyond this road boundary of the city proper. there lie extensive suburbs, the chief of which are Chitpore on the north, Nunden Baugh, BaharSimleah, Sealdah, Entally, and Ballygunge on the east, and Bhowaneepore, Allipore, and Kidderpore or on the south. The villages of Sulkeah, Howra, The water supply of C. has recently been very |