Evanescent Increments and Unnecessary Perplexities

Newton's calculus relied on 'evanescent increments' or infinitely small numbers, which critics like Berkeley found logically problematic. The trouble traces back to Zeno's paradox, which argues that motion is impossible because one must first cover half the distance, then half the remainder, ad infinitum. Numerically, the finite partial sums of 1/2 + 1/4 + 1/8 + ... never reach 1, though they approach it arbitrarily closely. This connects to the question of why 0.999... equals 1, an equality that can feel counterintuitive. Algebraic tricks (multiplying 0.333... by 3, or multiplying 0.999... by 10 and subtracting) seem to prove equality, but similar manipulations with divergent series like 1+2+4+8+... (which 'equals' -1) or Grandi's series 1-1+1-1+... (which can be made equal to 0, 1, or 1/2) show the danger. The solution, due to Cauchy, is to define the value of an infinite sum via the concept of a limit: the sum of a series is the number that the partial sums approach arbitrarily closely. Under this definition, 0.999... = 1, and Zeno's paradox is resolved because the infinite sum equals 1. Cauchy's rigorous approach avoids infinitesimals, but nonstandard analysis (Robinson) offers an alternative using actual infinitesimals. The key lesson: mathematical symbols do not have inherent meaning until we assign one, and good definitions avoid contradictions. Cauchy helped develop rigorous foundations for calculus, while later mathematicians refined them further. He faced resistance from colleagues and students at the École Polytechnique.