 An infinitesimal is a number smaller in absolute magnitude than any positive finite number, and yet not zero. Infinitesimals have had a chequered history. Early work in the calculus, as we have seen when presenting the structure of L’Hôpital’s 1696 treatise, was mostly based on a geometrical or kinematic (i.e. based on motion) understanding of infinitesimals. Here we treat infinitesimals arithmetically, i.e. within the context of the real number system, while conveying the key features of how the early analysts made use of them. To figure out the slope of the function \(f(x)=x^2\) precisely at one point, one considered an “infinitesimally small number” \(\epsilon\) and considered the slope of the straight line through \(f(x)\) and \(f(x+\epsilon)\).