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dkotschessaa
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I'm currently re-learning on my pre-calculus and calculus and have been reading Sylvanus P Thompson's _Calculus Made Easy_.
I am trying to figure out the place of this book within the framework of mathematics in general. He tells you he is going to teach you "those beautiful methods which are generally called by the terrifying names of the Differential Calculus and the Integral Calculus."
He does indeed make this look very easy, and I'm able to follow along in the book by simply reading along. But it occurred to me at some point that he has left out a few things that I remember from calculus (about 13 years ago), and am made to understand that this is a different approach.
According to http://en.wikipedia.org/wiki/Calculus_Made_Easy" (the completely accurate source of information for everything in the world) "Calculus Made Easy ignores the use of limits with its epsilon-delta definition, replacing it with a method of approximation directly to the correct answer in the spirit of Leibniz, now formally justified in modern nonstandard analysis."
Could somebody first explain to me in simpler terms what this means?
Later it is explained on the http://en.wikipedia.org/wiki/Leibniz#Calculus" that "Leibniz is credited, along with Sir Isaac Newton, with the inventing of infinitesimal calculus (that comprises differential and integral calculus)."
On the page on http://en.wikipedia.org/wiki/Infinitesimal_calculus" it is then remarked that "In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Bishop Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 1734."
Am I to assume that this is referring to the fact that Mr. Thompson has us discard those parts of the equation which he regards as "infinitesimally minute?" I did find this troubling when I first read the book - though of course one winds up with a perfectly reasonable answer - and can someone explain this in regards to the use of limits as another approach?
Thanks,
Dave KA
I am trying to figure out the place of this book within the framework of mathematics in general. He tells you he is going to teach you "those beautiful methods which are generally called by the terrifying names of the Differential Calculus and the Integral Calculus."
He does indeed make this look very easy, and I'm able to follow along in the book by simply reading along. But it occurred to me at some point that he has left out a few things that I remember from calculus (about 13 years ago), and am made to understand that this is a different approach.
According to http://en.wikipedia.org/wiki/Calculus_Made_Easy" (the completely accurate source of information for everything in the world) "Calculus Made Easy ignores the use of limits with its epsilon-delta definition, replacing it with a method of approximation directly to the correct answer in the spirit of Leibniz, now formally justified in modern nonstandard analysis."
Could somebody first explain to me in simpler terms what this means?
Later it is explained on the http://en.wikipedia.org/wiki/Leibniz#Calculus" that "Leibniz is credited, along with Sir Isaac Newton, with the inventing of infinitesimal calculus (that comprises differential and integral calculus)."
On the page on http://en.wikipedia.org/wiki/Infinitesimal_calculus" it is then remarked that "In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Bishop Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 1734."
Am I to assume that this is referring to the fact that Mr. Thompson has us discard those parts of the equation which he regards as "infinitesimally minute?" I did find this troubling when I first read the book - though of course one winds up with a perfectly reasonable answer - and can someone explain this in regards to the use of limits as another approach?
Thanks,
Dave KA
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