CALCULUS CALCUTTA 571 dently of each other, invented the differential calculus, although differing in the form in , which they conceived of and expressed the same truths. Newton's discovery or invention was made in 1665, and that of Leibnitz several years later. The notation of the latter was so con- venient, and his mode of attacking the subject has such a practical superiority for the learner, that Newton's method of fluxions has no w gone completely out of use ; although in a metaphys- ical point of view Newton's mode is not open to the objections which may be brought against that of Leibnitz. The discovery of this method originated in the investigation of curved lines, but is extended to the consideration of every species ol magnitude. Newton conceived of a curved line as generated by the motion of a point ; and the spirit of his method consists in determining the velocity with which the point, at each instant, is moving in a given direction different from that of the line; that is, e. g., if the point be moving in a general southwesterly direction, in determining the velocity with which it souths compared with that with which it wests. The spirit of Leibnitz's method con- sists in supposing the curve to be composed of infinitely short straight lines, and in determin- ing the direction of each of these lines. La- grange in his Thenrie des fonctions endeavored to treat the calculus from a purely algebraic point of view, and invented a new notation, but in his other works he always made use of the notation of Leibnitz. The INTEGRAL CAL- CULUS is the reverse of the differential, and seeks to find from a known ratio between the changes of two quantities mutually dependent on each other what the relation or law of de- pendence between the quantities themselves must be ; or, in the language of the calculus, the integral of a given function (i. e., law of dependence) is a required new function of which the given function is the differential. The CALCULUS OF VARIATIONS investigates the changes produced by gradually altering the laws of dependence which bind the variable quantities together. This invention of La- grange crowns the calculus of functions, which by means of these five branches is capable, un- der a master's hand, of tracing out very com- plicated and intricate chains of inter-depen- dence in every part of the domain of quantity. And yet there is not one of these calculi that can answer all the questions which the physical sciences ask of it. More powerful engines of analysis may yet be invented by future mathe- maticians. The CALCULUS OF QUATERNIONS, published by Sir W. R. Hamilton in 1853, promises to do something toward supplying this defect. By combining in one notation the direction as well as the length of a line, he is able to express in a single symbolical sentence an amount of geometrical truth which in ordi- nary analytical geometry would require at least four sentences. No other writer has yet mas- tered this powerful instrument sufficiently to use it with ease ; but the verdict of mathema- ticians is unanimous in praise of its ingenuity and probable future utility. The difference be- tween the powers of the principal calculi may be familiarly illustrated by the cycloid, a curve described by a nail head in the tire of a wheel rolling on a straight level road. The differen- tial calculus would investigate the direction in which the nail head moves at each instant of its motion, and show the proportion between its rise, its fall, its horizontal motion, its motion through space, the curvature of its real path, and the revolution of the wheel at each instant. The integral calculus would, from these ele- ments, discover how far the nail head travelled in one revolution of the wheel, how much space is enclosed between its path and the ground, &c. The calculus of variations would con- sider the change made by the wheel rolling over a hill, or would show how the cycloid differs in its properties from similar curves. The calculus is too difficult and abstruse for any popular exposition. The reader may find gen- eral views upon the subject in Davies's " Logic of Mathematics," and Comte's "Philosophy of Mathematics," translated by Prof. Gillespie, or in French in Carnot's Reflexion*. For gaining a practical acquaintance with the science there are numerous accessible treatises, among which Church's and Courtenay's are well adapted to ordinary students, but Peirce's conducts much more rapidly into the highest walks. Of Eng- lish treatises, Price's holds a high rank; but the most extensive treatise in the English lan- guage is that by Augustus De Morgan, pub- lished by the society for the diffusion of useful knowledge. The treatise of I. Todhunter is highly esteemed as a practical work. Among the best German works is that of Dr. Martin Ohm. The French have been prolific writers upon the subject ; among them Duhamel holds a high rank, and the treatise of Lacroix (3 vols. 4to, 1810-'19) is the most elaborate that has yet appeared in any language. CALCUTTA (Kali Ghatta, the ghaut or land- ing place of the goddess Kali, wife of Siva), a city of Hindostan, capital of the province of Bengal, metropolis of British India, and seat of the supreme government of that country, sit- uated 100 m. from the sea, on the E. bank of the Hoogly river, the W. branch of the Ganges, its citadel being in lat. 22 34' 49" N., Ion. 88 27' 16" E. In 1866 the population of the city proper was 377,924, of whom 239,190 were Hindoos, 113,059 Mohammedans, 11,224 Euro- peans, 11,036 Eurasians (progeny of a Euro- pean father and a native mother), and 681 Jews ; the population of the suburbs was 238,- 325; total of city and suburbs, 616,249. The population of the city in 1872 was 447, 601. On ascending the Hoogly, the scenery, which for many miles from the sea is dreary and unin- viting, becomes more picturesque as one ap- proaches Calcutta. No land is visible at the mouth of the river, the channel of which is marked out by lighthouses and buoys, and must be followed many miles inland before the
