infinitesimal,in mathematics, a quantity less than any finite quantity , yet not zero. Even though no such quantity can exist in the real number system, many early attempts at the to justify calculus were based on sometimes dubious reasoning about infinitesimals: derivatives were defined as ultimate ratios of infinitesimals, and integrals were calculated by summing rectangles of infinitesimal width. As a result, differential and integral calculus was originally referred to as the infinitesimal calculus. This terminology gradually disappeared as the rigorous concepts of limit, continuity, and the real numbers became clarifiedwere formulated.