Mathematics in Ancient and Medieval India c9.10

c9.10

( a ) *Modern concept*—The differential calculus is directly concerned with the momentary state of a phenomenon. This momentary state is symbolised by the differential coefficient. In order to render different processes such as change of position, change of motion, of temperature, volume etc. susceptible to mathematical treatment, the magnitude are supposed to change during a series of very short intervals of time. The shorter the interval, the more uniform is the process. The conception is of fundamental importance. The velocity of a train, sixty miles per hour, does not mean that the train can run one mile per minute. The speed of the train might vary in successive minutes, the sum total of which for sixty successive minutes or one hour amount to sixty miles. Hence, the average velocity of the train is one mile per minute. If we continue to reduce the unit of time, the average velocity approaches more and more nearly to the actual velocity of the train during the whole time under consideration. If x₁ be the distance traversed at the time t₁ when the observation commences and x₂ be the distance traversed at the time t₂ then the velocity of the train, v = (x₂ - x₁)/(t₂ - t₁). If t₂ - t₁ be extremely small then (x₂ - x₁)/(t₂ - t₁) may be represented as ()/() where denotes the space traversed in the small unit of time δ t with uniform motion.