Pebble-counting Latin name became physics' backbone
Calculus studies continuous change and underwrites modern analysis—Newton and Leibniz forging infinitesimal versions in the late 1600s before limits cleaned the foundations. Latin calculus means pebble, recalling Roman counters. Differential and integral halves let slopes and areas speak about curves.
The pebble sense survives in medicine, where a calculus is a stone. Romans reckoned by setting pebbles on counting boards, so doing sums was placing pebbles, and settling a debt meant calling someone to the pebbles. The verb calculare shows up in Spain around 400 AD, and English used calculate by 1672, a few years before Newton and Leibniz published their Latin texts. The word also names other formal systems, from lambda and propositional calculus to Bentham's felicific calculus of pleasure.
The first way of handling tiny quantities was the infinitesimal, a number larger than zero yet smaller than every term of 1, 1/2, 1/3 and so on. Because nobody could make that idea precise, the late 19th century replaced it with the epsilon-delta treatment of limits, which became standard in the 20th century. Infinitesimals returned later through non-standard analysis.
A straight line's slope is rise over run, but a curve's slope keeps changing. The derivative takes the slope of a secant line through two nearby points and lets their gap shrink toward zero, sidestepping division by zero; the result is the slope of the tangent line. If the input is time and the output a ball's position, the derivative is its velocity. Lagrange wrote f prime, so squaring x gives 2x, while Leibniz's dy over dx was meant as a ratio of two infinitely small changes.
Integration runs the other way. At a steady 50 miles per hour for three hours, distance is just the 150-mile area of a rectangle under the speed graph. When speed varies, one slices time into short intervals, adds up the rectangles in a Riemann sum, and takes the limit as the slices thin out. The indefinite integral, or antiderivative, reverses differentiation, and the fundamental theorem ties the two branches together.
Source: Calculus
