primarily denotes a little stone or pebble, anciently used in making computations, taking of suffrages, playing at tables, and the like. In after times, pieces of ivory, and counters struck of silver, gold, and other matters, were used in lieu thereof, but still retaining the ancient names. Computists were by the lawyers called calculones, when they were either slaves, or newly freed men; those of a better condition were named calculatores; or numerarii: ordinarily there was one of these in each family of distinction. The Roman judges anciently gave their opinions by calculi, which were white for abolition, and black for condemnation. Hence calculus albus, in ancient writers, denotes a favourable vote, either in a person to be absolved and acquitted of a charge, or elected to some dignity or post; as calculus niger did the contrary. This usage is said to have been borrowed from the Thracians, who marked their happy or prosperous days by white, and their unhappy by black, pebbles, put each night into an urn.
Besides the diversity of colour, there were some calculi also which had figures or characters engraven on them, as those which were in use in taking the suffrages both in the senate and at assemblies of the people. These calculi were made of thin wood, polished and covered over with wax. Their form is still seen in some medals of the Caecilian family; and the manner of casting them into the urns, in the medals of the Licinian family. The letters marked upon these calculi were U. R. for uti rogas, and A. for antiquo; the first of which expressed an approbation of the law, the latter a rejection of it. Afterwards the judges who sat in capital causes used calculi marked with the letter A. for absolutio; C. for condemnatio; and N. L. for non luet, signifying that a more full information was required.
Calculus is also used in ancient geometric writers for a kind of weight equal to two grains of cicer. Some make it equivalent to the siliqua, which is equal to three grains of barley. Two calculi made the centarium.
Mathematics, is a certain method of performing investigations and resolutions, particularly in mechanical philosophy. Thus there is the Differential calculus, the Exponential, the Integral, the Literal, and the Antecedential.
Calculus Differentialis, is a method of differencing quantities, or of finding an infinitely small quantity, which being taken infinite times, shall be equal to a given quantity; or, it is the arithmetic of the infinitely small differences of variable quantities.
The foundation of this calculus is an infinitely small quantity, or an infinitesimal, which is a portion of a quantity incomparable to that quantity, or that is less than any assignable one, and therefore accounted as nothing; the error accruing by omitting it being less than any assignable one. Hence two quantities, only differing by an infinitesimal, are reputed equal. Thus, in astronomy, the diameter of the earth is an infinitesimal, in respect of the distance of the fixed stars; and the same holds in abstract quantities. The term, infinitesimal, therefore, is merely relative, and involves a relation to another quantity; and does not denote any real ens or being. Now infinitesimals are called differentials, or differential quantities, when they are considered as the differences of two quantities. Sir Isaac Newton calls them moments; considering them as the momentary increments of quantities, e.g., of a line generated by the flux of a point, or of a surface by the flux of a line. The differential calculus, therefore, and the doctrine of fluxions, are the same thing under different names; the former given by M. Leibnitz, and the latter by Sir Isaac Newton: each of whom lays claim to the discovery. There is, indeed, a difference in the matter of expressing the quantities resulting from the different views wherein the two authors consider the infinitesimals: the one as moments, the other as differences. Leibnitz, and most foreigners, express the differentials of quantities by the same letters as variable ones, only prefixing the letter \(d\): thus the differential of \(x\) is called \(dx\); and that of \(y\), \(dy\); now \(dx\) is a positive quantity, if \(x\) continually increase; negative, if it decrease. The English, with Sir Isaac Newton, Calculus. Newton, instead of \( dx \) write \( x \) (with a dot over it), for \( dy \), \( y \), &c., which foreigners object against, on account of that confusion of points, which they imagine arises when differentials are again differenced; besides, that the printers are more apt to overlook a point than a letter. Stable quantities being always expressed by the first letters of the alphabet \( d^a = o \), \( d^b = o \), \( d^c = o \); therefore \( d(x+y-a) = dx + dy \), and \( d(x-y+a) = dx - dy \). So that the differencing of quantities is easily performed by the addition or subtraction of their compounds.
To difference quantities that multiply each other; the rule is, first, multiply the differential of one factor into the other factor, the sum of the two factors is the differential sought: thus, the quantities being \( x \), \( y \), the differential will be \( x \cdot dy + y \cdot dx \), i.e. \( d(xy) = xy \cdot dx + y \cdot dx \).
Secondly, If there be three quantities mutually multiplying each other, the factum of the two must then be multiplied into the differential of the third; thus suppose \( vxy \), let \( vx = t \), then \( vx \cdot y = ty \); consequently \( d(vxy) = tdxy + y \cdot dt \); but \( dt = 0 \cdot dx + x \cdot dv \).
These values, therefore, being substituted in the antecedent differential, \( tdxy + y \cdot dt \), the result is, \( d(vxy) = v \cdot dx + v \cdot dy + x \cdot dv \). Hence it is easy to apprehend how to proceed, where the quantities are more than three. If one variable quantity increase, while the other decreases, it is evident \( y \cdot dx - x \cdot dy \) will be the differential of \( xy \).
To difference quantities that mutually divide each other; the rule is, first, multiply the differential of the divisor into the dividend; and on the contrary, the differential of the dividend into the divisor; subtract the last product from the first, and divide the remainder by the square of the divisor; the quotient is the differential of the quantities mutually dividing each other. See Fluxions.
Calculus Exponentialis, is a method of differencing exponential quantities, or of finding and summing up the differentials or moments of exponential quantities; or at least bringing them to geometrical constructions.
By exponential quantity, is here understood a power, whose exponent is variable; e.g. \( x^a \), \( a^x \), \( x^7 \), where the exponent \( x \) does not denote the same in all the points of a curve, but in some stands for 2, in others for 3, in others for 5, &c.
To difference an exponential quantity; there is nothing required but to reduce the exponential quantities to logarithmic ones; which done, the differencing is managed as in logarithmic quantities.—Thus, suppose the differential of the exponential quantity \( x^y \) required, let
\[ x^y = z \]
Then will \( y \cdot dx = dz \)
\[ \frac{1}{x} \cdot dy + \frac{y}{x} \cdot dx = \frac{dz}{z} \]
That is, \( x^y \cdot dy + y \cdot x^{y-1} \cdot dx = dz \).
Calculus Integralis, or Summatorius, is a method of integrating, or summing up moments, or differential quantities; i.e. from a differential quantity given, to find the quantity from whose differencing the given differential results.
The integral calculus, therefore, is the inverse of the differential one; whence the English, who usually call the differential method fluxions, give this calculus, which ascends from the fluxions, to the flowing or variable quantities; or as foreigners express it, from the differences to the sums, by the name of the inverse method of fluxions.
Hence, the integration is known to be justly performed, if the quantity found, according to the rules of the differential calculus, being differenced, produce that proposed to be summed.
Suppose \( f \) the sign of the sum, or integral quantity, then \( f(y \cdot dx) \) will denote the sum, or integral of the differential \( y \cdot dx \).
To integrate, or sum up a differential quantity: it is demonstrated, first, that \( f(x) = x \); secondly, \( f(d(x+dy)) = x + y \); thirdly, \( f(x \cdot dy + y \cdot dx) = x \cdot y \); fourthly, \( f(m \cdot x^m \cdot dx) = x^m : \) fifthly, \( f(n \cdot m \cdot x^m \cdot dx) = \frac{x^n}{m} \);
Sixthly, \( f(y \cdot dx - x \cdot dy) : y^2 = x \cdot y \). Of these, the fourth and fifth cases are the most frequent, wherein the differential quantity is integrated, by adding a variable unity to the exponent, and dividing the sum by the new exponent multiplied into the differential of the root; v.g. the fourth case, by \( m - (i + 1) \cdot dx \), i.e. by \( m \cdot dx \).
If the differential quantity to be integrated doth not come under any of these formulas, it must either be reduced to an integral finite, or an infinite series, each of whose terms may be summed.
It may be here observed, that, as in the analysis of finites, any quantity may be raised to any degree of power; but vice versa, the root cannot be extracted out of any number required; so in the analysis of infinites, any variable or flowing quantity may be differenced; but vice versa, any differential cannot be integrated. And as, in the analysis of finites, we are not yet arrived at a method of extracting the roots of all equations, so neither has the integral calculus arrived at its perfection: and as in the former we are obliged to have recourse to approximation, so in the latter we have recourse to infinite series, where we cannot attain to a perfect integration.
Calculus Literalis, or Literal Calculus, is the same with specious arithmetic, or algebra, so called from its using the letters of the alphabet; in contradistinction to numeral arithmetic, which uses figures. In the literal calculus given quantities are expressed by the first letters, \( a \), \( b \), \( c \), \( d \); and quantities sought by the last \( y \), \( z \), &c. Equal quantities are denoted by the same letters.
Calculus, Antecedental, a geometrical method of reasoning invented by Mr Glenie, which, without any consideration of motion or velocity, is applicable to all the purposes of fluxions. In this method, says Mr Glenie, "every expression is truly and strictly geometrical, is founded on principles frequently made use of by the ancient geometers, principles admitted into the very first elements of geometry, and repeatedly used by Euclid himself. As it is a branch of general geometrical proportion, or universal comparison, and is derived from an examination of the antecedents of ratios, hav- Calculating given consequents and a given standard of comparison in various degrees of augmentation and diminution they undergo by composition and decomposition, I have called it the antecedental calculus. As it is purely geometrical, and perfectly scientific, I have since its first occurrence to me in 1779, always made use of it instead of the fluxionary and differential calculi, which are merely arithmetical. Its principles are totally unconnected with the ideas of motion and time, which, strictly speaking, are foreign to pure geometry and abstract science, though, in mixed mathematics and natural philosophy, they are equally applicable to every investigation, involving the consideration of either with the two numerical methods just mentioned. And as many such investigations require compositions and decompositions of ratios, extending greatly beyond the triplicate and subtriplicate, this calculus in all of them furnishes every expression in a strictly geometrical form. The standards of comparison in it may be any magnitudes whatever, and are of course indefinite and innumerable; and the consequents of the ratios, compound or decompounded, may be either equal or unequal, homogeneous or heterogeneous. In the fluxionary and differential methods, on the other hand, 1, or unit, is not only the standard of comparison, but also the consequent of every ratio compounded or decompounded." See Phil. Trans. Edin. vol. iv.
Some mathematicians, however, are of opinion that the advantage to be derived from the employment of this calculus is not so great as the author seems to promise from it.
**Calculus Minerve**, among the ancient lawyers, denoted the decision of a cause, wherein the judges were equally divided. The expression is taken from the history of Oracles, represented by Aeschylus and Euripides; at whose trial, before the Areopagites, for the murder of his mother, the votes being equally divided for and against him, Minerva interposed, and gave the casting vote or calculus in his behalf.
M. Cramer, professor at Marburg, has a discourse exprefs, De Calculo Minerve; wherein he maintains, that all the effect an entire equality of voices can have, is to leave the cause in statu quo.
**Calculus Tiburtinus**, a sort of figured stone, formed in great plenty about the catacombs of the Anio, and other rivers in Italy; of a white colour, and in shape oblong, round, or echinated. They are a species of the *floria lapidea*, or *flaletites*, and generated like them; and to like fugar plums, that it is a common jest at Rome to deceive the unexperienced by serving them up as desserts.
**Calculus**, in Medicine, the discale of the stone in the bladder, or kidneys. The term is Latin, and signifies a little pebble. The calculus in the bladder is called lithiasis; and in the kidneys, nephritis. See Medicine and Surgery.
Human calculi are commonly formed of different strata or incrustations; sometimes smooth and heavy like mineral stones; but often rough, spongy, light, and full of inequalities or protuberances: chemically analyzed, or distilled in an open fire, they nearly yield the same principles as urine itself, or at least an empyreumatic volatile urinous matter, together with a great deal of air. They never have, nor can have, naturally, any foreign matter for a basis; but they may by accident; an instance of which is related by Dr. Calcutta, Percival*. A bougie had unfortunately slipped into the bladder, and upon it a stone of considerable size was formed in less than a year. This stone had so much the appearance of chalk, that the doctor was induced to try whether it could be converted into quicklime. His experiment succeeded, both with that and some other calculi; from which he conjectures, that hard waters which contain calcareous earth may contribute towards the formation of these calculi.
**Calcutta**, the capital of the province of Bengal, and of all the British possessions in the East Indies, is situated on the river Huguley, a branch of the Ganges, about 100 miles from the sea, in N. Lat. 23° and Long. 88° 28' E. from Greenwich. It is but a modern city, built on the site of a village called Gouripur. The English first obtained the Mogul's permission to settle in this place in the year 1692; and Mr Job Charnock, the company's agent, made choice of the spot on which the city stands, on account of a large shady grove which grew there; though in other respects it was the worst he could have pitched upon; for three miles to the north coast, there is a salt water lake, which overflows in September, and when the flood retires in December, leaves behind such a quantity of fish and other putrefied matter, as renders the air very unhealthy. The custom of the Gentoos throwing the dead bodies of their poor people into the river is also very disagreeable, and undoubtedly contributes to render the place unhealthy, as well as the cause already mentioned.
Calcutta is now become a large and populous city, being supposed at present to contain 500,000 inhabitants. It is elegantly built, at least the part inhabited by the English; but the rest, and that the greatest part, is built after the fashion of the cities of India in general. The plan of all these is nearly the same; their streets are exceedingly confined, narrow, and crooked, with a vast number of ponds, reservoirs, and gardens interspersed. A few of the streets are paved with brick. The houses are built, some with brick, others with mud, and a still greater number with bamboos and mats; all which different kinds of fabrics standing intermixed with one another, form a very uncouth appearance. The brick houses are seldom above two stories high, but those of mud and bamboos are only one, and are covered with thatch. The roofs of the brick houses are flat and terraced. These, however, are much fewer in number than the other two kinds; so that fires, which often happen, do not sometimes meet with a brick house to obstruct their progress in a whole street. Within these 20 or 25 years Calcutta has been greatly improved both in appearance and in the salubrity of its air: the streets have been properly drained, and the ponds filled; thereby removing a vast surface of stagnant water, the exhalations of which were particularly hurtful. The citadel is named Fort William, and is superior as a fortress to any in India; but is now on too extensive a scale to answer the purpose for which it was intended, viz. the holding a post in case of extremity. It was begun on this extended plan by Lord Clive immediately after the battle of Plassey. The expense attending it was supposed to amount to two millions sterling.
Calcutta is the emporium of Bengal, and the residence dence of the governor general of India. Its flourishing state may in a great measure be supposed owing to the unlimited toleration of all religions allowed here; the Pagans being suffered to carry their idols in procession, the Mahomedans not being discomfited, and the Roman Catholics being allowed a church.
At about a mile's distance from the town is a plain where the natives annually undergo a very strange kind of penance on the 9th of April; some for the sins they have committed, others for those they may commit, and others in consequence of a vow made by their parents. This ceremony is performed in the following manner: Thirty bamboos, each about the height of 20 feet, are erected in the plain above-mentioned. On the top of these they contrive to fix a swivel, and another bamboo of thirty feet or more crosses it, at each end of which hangs a rope. The people pull down one end of this rope, and the devotee placing himself under it, the brahmin pinches up a large piece of skin under both the shoulderblades, sometimes in the breasts, and thrusts a strong iron hook through each. These hooks have lines of Indian garlands hanging to them, which the priest makes fast to the rope at the end of the cross bamboo, and at the same time puts a fish round the body of the devotee, laying it loyally in the hollow of the hooks, left by the skin's giving way, he should fall to the ground. When this is done, the people haul down the other end of the bamboo; by which means the devotee is immediately lifted up 30 feet or more from the ground, and they run round as fast as their legs can carry them. Thus the devotee is thrown out the whole length of the rope, where, as he swings, he plays a thousand antic tricks; being painted and dressed in a very particular manner, on purpose to make him look more ridiculous. Some of them continue swinging half an hour, others less. The devotees undergo a preparation of four days for this ceremony. On the first and third they abstain from all kinds of food; but eat fruit on the other two. During this time of preparation they walk about the streets in their fantastic dresses, dancing to the sound of drums and horns; and some to express the greater ardour of devotion, run a rod of iron quite through their tongues, and sometimes through their cheeks also.
Before the war of 1755, Calcutta was commonly garrisoned by 300 Europeans, who were frequently employed in conveying the company's vessels from Patna, loaded with saltpetre, piece goods, opium, and raw silk. The trade of Bengal alone supplied rich cargoes for 50 or 60 ships annually, besides what was carried on in small vessels to the adjacent countries. It was this flourishing state of Calcutta that probably was one motive for the nabob Surajah Dowlah to attack it in the year 1756. Having had the fort of Cossimbazar delivered up to him, he marched against Calcutta with all his forces, amounting to 70,000 horse and foot, with 400 elephants, and invested the place on the 15th of June. Previous to any hostilities, however, he wrote a letter to Mr Drake the governor, offering to withdraw his troops, on condition that he would pay him his duty on the trade for 15 years past, defray the expense of his army, and deliver up the black merchants who were in the fort. This being refused, he attacked one of the redoubts at the entrance of the town; but was repulsed with great slaughter. On the 16th he attacked another advanced post, but was likewise repulsed with great loss. Notwithstanding this disappointment, however, the attempt was renewed on the 18th, when the troops abandoned these posts, and retreated into the fort; on which the nabob's troops entered the town, and plundered it for 24 hours. An order was then given for attacking the fort; for which purpose a small breastwork was thrown up, and two twelve-pounders mounted upon it; but without firing oftener than two or three times an hour. The governor then called a council of war, when the captain of the train informed them, that there was not ammunition in the fort to serve three days; in consequence of which the principal ladies were sent on board the ships lying before the fort. They were followed by the governor, who declared himself a Quaker, and left the place to be defended by Mr Holwell the second in council. Besides the governor, four of the council, eight gentlemen of the company's service, four officers, and 100 soldiers, with 52 free merchants, captains of ships, and other gentlemen, escaped on board the ships, where were also 59 ladies, with 33 of their children. The whole number left in the fort was about 250 effective men, with Mr Holwell, four captains, five lieutenants, six ensigns, and five sergeants; as also 14 sea captains, and 29 gentlemen of the factory. Mr Holwell then having held a council of war, divided three chests of treasure among the discontented soldiers, making them large promises also, if they behaved with courage and fidelity; after which he boldly stood on the defence of the place, notwithstanding the immense force which opposed him. The attack was very vigorous; the enemy having got possession of the houses, called the English from thence, and drove them from the battions; but they themselves were several times dislodged by the fire from the fort, which killed an incredible number, with the loss of only five English soldiers the first day. The attack, however, was continued till the afternoon of the 20th; when many of the garrison being killed and wounded, and their ammunition almost exhausted, a flag of truce was hung out. Mr Holwell intended to have availed himself of this opportunity to make his escape on board the ships, but they had fallen several miles down from the fort, without leaving even a single boat to facilitate the escape of those who remained. In the mean time, however, the back-gate was betrayed by the Dutch guard, and the enemy, entering the fort, killed all they first met, and took the rest prisoners.
The fort was taken before six in the evening; and, in an hour after, Mr Holwell had three audiences of the nabob, the last being in the durbar or council. In all of these the governor had the most positive assurances that no harm should happen to any of the prisoners; but he was surprized and enraged at finding only 500l. in the fort, instead of the immense treasures he expected; and to this as well as perhaps to the resentment of the jemidaars or officers, of whom many were killed in the siege, we may impute the catastrophe that followed.
As soon as it was dark, the English prisoners, to the number of 146, were directed by the jemidaars who guarded them, to collect themselves into one body, and sit down quietly under the arched veranda, Besides the guard over them, another was placed at the south end of this veranda, to prevent the escape of any of them. About 500 gunmen, with lighted matches, were drawn up on the parade; and soon after the factory was in flames to the right and left of the prisoners, who had various conjectures on this appearance. The fire advanced with rapidity on both sides; and it was the prevailing opinion of the English, that they were to be suffocated between the two fires. On this they soon came to a resolution of rushing on the guard, seizing their scimitars, and attacking the troops upon the parade, rather than be thus tamely roasted to death: but Mr Holwell advanced, and found the Moors were only searching for a place to confine them in. At the time Mr Holwell might have made his escape, by the affluence of Mr Leech, the company's smith, who had escaped when the Moors entered the fort, and returned just as it was dark, to tell Mr Holwell he had provided a boat, and would ensure his escape, if he would follow him through a passage few were acquainted with, and by which he then entered. This might easily have been accomplished, as the guard took little notice of it; but Mr Holwell told Mr Leech, he was resolved to share the fate of the gentlemen and the garrison; to which Mr Leech gallantly replied, that "then he was resolved to share Mr Holwell's fate, and would not leave him."
The guard on the parade advanced, and ordered them all to rise and go into the barracks. Then, with their muskets presented, they ordered them to go into the Black Hole prison; while others, with clubs and scimitars, pressed upon them so strong, that there was no resisting it; but, like an agitated wave impelling another, they were obliged to give way and enter; the rest following like a torrent. Few among them, the foldiers excepted, had the least idea of the dimensions or nature of a place they had never seen; for if they had, they should at all events have rushed upon the guard, and been cut to pieces by their own choice as the lesser evil.
It was about eight o'clock when these 146 unhappy persons, exhausted by continual action and fatigue, were thus crammed together into a dungeon about eighteen feet square, in a close, filthy night in Bengal; shut up to the east and south, the only quarters from whence air could reach them, by dead walls, and by a wall and door to the north; open only to the west by two windows, strongly barred with iron, from which they could receive scarce any circulation of fresh air.
They had been but few minutes confined before every one fell into a perspiration so profuse, that no idea can be formed of it. This brought on a raging thirst, which increased in proportion as the body was drained of its moisture. Various expedients were thought of to give more room and air. Every man was stripped, and every hat put in motion; they several times sat down on their hams; but at each time several of the poor creatures fell, and were instantly suffocated or trode to death.
Before nine o'clock every man's thirst grew intolerable, and respiration difficult. Efforts were again made to force the door; but still in vain. Many insults were used to the guards, to provoke them to fire in upon the prisoners, who grew outrageous, and many delirious. "Water, water," became the general cry. Some water was brought; but these supplies, like sprinkling water on fire, only served to raise and feed the flames. The confusion became general and horrid from the cries and ravings for water; and some were trampled to death. This scene of misery proved entertainment to the brutal wretches without, who supplied them with water, that they might have the satisfaction of seeing them fight for it, as they phrased it; and held up lights to the bars, that they might lose no part of the inhuman diversion.
Before eleven o'clock, most of the gentlemen were dead, and one-third of the whole. Thirst grew intolerable; but Mr Holwell kept his mouth moist by sucking the perspiration out of his shirt sleeves, and catching the drops as they fell, like heavy rain, from his head and face. By half an hour after eleven, most of the living were in an outrageous delirium. They found that water heightened their uneasiness; and "Air, air" was the general cry. Every insult that could be devised against the guard; all the opprobrious names that the viceroy and his officers could be loaded with, were repeated, to provoke the guard to fire upon them. Every man had eager hopes of meeting the first shot. Then a general prayer to heaven, to hasten the approach of the flames to the right and left of them, and put a period to their misery. Some expired on others; while a steam arose as well from the living as the dead, which was very offensive.
About two in the morning, they crowded so much to the windows, that many died standing, unable to fall by the throng and equal pressure round. When the day broke, the stench arising from the dead bodies was insufferable. At that juncture, the fouubah, who had received an account of the havoc death had made among them, sent one of his officers to inquire if the chief survived. Mr Holwell was shown to him; and near six, an order came for their release.
Thus they had remained in this infernal prison from eight at night until six in the morning, when the poor remains of 146 souls, being only 23, came out alive; but most of them in a high putrid fever. The dead bodies were dragged out of the hole by the foldiers, and thrown promiscuously into the ditch of an unfinished ravelin, which was afterwards filled with earth.
The injuries which Calcutta suffered at this time, however, were soon repaired. The place was retaken by Admiral Watson and Colonel Clive, early in 1757; Surajah Dowla was defeated, depopulated, and put to death; and Meer Jaffier, who succeeded him in the nabobship, engaged to pay an immense sum for the indemnification of the inhabitants. Since that time, the immense acquisition of territory by the British in this part of the world, with the constant state of security enjoyed by this city, have given an opportunity of embellishing and improving it greatly beyond what it was before.—Among these improvements we may reckon that of Sir William Jones, who, on the 15th of January 1784, instituted a society for inquiring into the history civil and natural, the antiquities, arts, sciences, and literature of Asia; and thus the literature