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 absolution, 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 engraved 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 absolvus; C. for condemnno; and N. L. for non liquet, signifying that a more full information was required.

Calculus is also used in ancient grammatic writers for a kind of weight equal to two grains of cicer. Some make it equivalent to the filiqua, which is equal to three grains of barley. Two calculi made the centrum.

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 retrospective, 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, v.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 lay claim to the discovery. There is, indeed, a difference in the manner 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, 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 \(da = o\), \(db = o\), \(dc = o\); 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 dif. differential will be \( x \cdot dy + y \cdot dx \), i.e., \( d(xy) = x \cdot dy + y \cdot dx \).

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 \( v = x \); then \( vx = y \); consequently \( d(vxy) = dy + y \cdot dx \); but \( d = -x \cdot dx + x \cdot dv \).

These values, therefore, being substituted in the antecedent differential, \( t \cdot dy + y \cdot dt \), the result is, \( d(vxy) = v \cdot dy + y \cdot dx + xy \cdot dv \). 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., \( x^a \cdot x^b \), 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 \( xy \) required, let

\[ \begin{align*} xy &= z \\ \text{Then will } y \cdot dx &= dz \\ l \cdot dy + \frac{y}{x} \cdot dx &= dz \\ z \cdot l \cdot dy + \frac{z}{x} \cdot dx &= dz \end{align*} \]

That is, \( xy \cdot l \cdot dy + xy^{-1} \cdot dx = dz \).

**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 \( f \cdot y \cdot dx \) 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, \( f(dx + dy) = x + y \); thirdly, \( f(x \cdot dy + y \cdot dx) = xy \); fourthly, \( f(m^{n-1} \cdot dx) = m^n \); fifthly, \( f(m^n \cdot dx) = \frac{n}{m} \cdot dx \); sixthly, \( f(y \cdot dx - x \cdot dy) = y = x \cdot y \). 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 \( m - (1 + 1) \cdot dx \), 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 contradiction 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 \( x y z \), &c. Equal quantities are denoted by the same letters.

**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 Orestes, 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 *furiae lapidea*, and generated like them; and so like fugar-plums in the whole, that it is a common jest at Rome to deceive the unexperienced by serving them up at feasts.

**Calculus**, in Medicine, the disease of the stone in the bladder, or kidneys. The term is Latin, and signifies a little pebble. The calculus in the bladder is called *litiasis*; and in the kidneys, *nephritis*. See Medicine and Surgery.

Human calculi are commonly formed of different flata