Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus

Decomposing the Peer Effect on Adolescent Substance Use: Mediation, Nonlinearity, and Differential Nonlinearity*

“…Moreover, saturation effects are not a discovery or specific to social learning theory. We expect causal mechanisms in general to have maximum points, in much the same way that we observe terminal velocity (Wilson, 1919); diminishing returns (see Schneider, 1964; Shephard, 1970); limits in mathematical functions (Felscher, 2000; Grabiner, 1983; Russ, 2004); simple and complex (e.g., the Kolmogorov model) “laws of population growth” (Brauer and Castillo‐Chávez, 2001: 196–9; Lotka, 1925: 64–76; Malthus, 1798: 1–11); and saturation wave models (Centers for Disease Control and Prevention [CDC], 1992) in epidemiology. For example, common‐source, continuous‐exposure, chronic‐disease endemic models are characterized by a gradual decrease in the rate of exponential growth of disease incidence until saturation is reached and a plateau is observed rather than a peak (CDC, 1992; see also Hox, 1995: 78).…”

Arc length of function graphs via Taylor's formula

“…In loc. cit., we show, using an idea of Poisson (see [2] or [6]), utilizing Taylor's formula with Lagrange remainder, that the following version of the FTC easily can be proved in just a few lines of calculation.…”

Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus

“…The book draws its title from an older essay, entitled Who Gave You the Epsilon? Cauchy and the origins of rigorous calculus [61]. The faint hope was that the book would approach the thesis implied by the title of the older essay, in a critical spirit, namely, as an ahistorical 1 tale in need of re-examination.…”

“…Generally speaking, Cauchy's limit concept is a kinetic one (see Hourya Benis Sinaceur [123]), and is a derived notion, depending for its definition on the primary notion of a variable quantity. On occasion, Cauchy worked with real inequalities in proofs (see, e.g., Grabiner [61]), but he never gave such a definition of continuity.…”

“…That happens all the time, of course. Our present day conception of Newton's differential calculus 61 The link provided is a link to an electronic version of Grabiner's article [61]. 62 See footnote 15.…”