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→Bernhard Riemann
That may sound like hair-splitting, but his approach could be rigorously formalized quite easily, while this was much more difficult for infinitesimals.--[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 10:36, 30 July 2018 (EDT)
That may sound like hair-splitting, but his approach could be rigorously formalized quite easily, while this was much more difficult for infinitesimals.--[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 10:36, 30 July 2018 (EDT)
:Not sure if there is a meaningful distinction here. Infinitely small is inherently a limit. But you make a very interesting point and I'll think about it further. Also, did Euler, Leibniz or Newton use the concept of infinitesimal?--[[User:Aschlafly|Andy Schlafly]] ([[User talk:Aschlafly|talk]]) 12:24, 30 July 2018 (EDT)
:Not sure if there is a meaningful distinction here. Infinitely small is inherently a limit. But you make a very interesting point and I'll think about it further. Also, did Euler, Leibniz or Newton use the concept of infinitesimal?--[[User:Aschlafly|Andy Schlafly]] ([[User talk:Aschlafly|talk]]) 12:24, 30 July 2018 (EDT)
#Fermat had already used infinitesimals to calculate the tangents of polynomials. For the parabola y=<math>x^2</math>, he would look at the secant through the points (x,<math>x^2</math>), (x+D,<math>(x+D)^2</math>) which are separated at the x-axis by the infinitesimal distance <math>D</math> and then calculate <math>slope = \frac{(x+D)^2 - x^2}{D} = </math> <math>\frac{2Dx+D^2}{D} = </math> <math>2x+D = </math> <math>2x</math> Calculations like this drove philosophers like Hobbes crazy: how can you say something is zero at one time and works not as zero another time. For us, it is easy to reconcile this by introducing limits, but those were not thought of for quite a while!
#Euler used infinitesimals creatively, Leibniz called them <math>dx</math> etc. If I recall correctly, Newton did not generally use them for integration: he transformed the function which he wanted to integrate into a power series, integrated the parts of the series (as it was well known how to integrate polynomials), and at last, tried to transform the resulting power-series back into a well-known function. The first and the third step often required quite an ingenuity
#In my opinion one of the great things about Riemann-Cauchy-integration is that it transformed an art into a craft: only few had the talent to use infinitesimals in a way that provided correct results!
--[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 15:06, 30 July 2018 (EDT)