When they came to require still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth's days, they called “second minùtes” (i.e.: small quantities of the second order of minuteness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called.
Calculus Made Easy by Silvanus P. Thompson (1910)
41–50 of 64 posts
Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#42Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus…
I have mixed feelings about this. I've been through nonstandard analysis. My response was, "We shouldn't need the axiom of choice to define the derivative." But I think that it is extremely important to understand that the infinitesimal notation really MEANS something. Here is some Python to demonstrate. # d is a functor. It takes a function and returns a second function. # The second function captures the change in…
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Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#43I love projects like this, but I really wish instead of converting text resources to web sites that these projects would produce epub outputs. It's great for distribution, offline reading, and scaling to different display sizes, aspect ratios, and resolutions.
The page links to a pdf version[0], which can easily be converted to epub using Calibre[0], which is free and open-source. [0] https://www.gutenberg.org/ebooks/33283 [1] https://calibre-ebook.com/
I'm familiar with those tools, but they don't preserve the formatting of the source document (when PDF) and certain types of embedded resources don't transfer properly.
Some PDF documents don't contain the source text as digitized text, either. It's just a bundle of scanned images.
Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#44I love projects like this, but I really wish instead of converting text resources to web sites that these projects would produce epub outputs. It's great for distribution, offline reading, and scaling to different display sizes, aspect ratios, and resolutions.
Epubs are horrible for technical documents
Only if they exist as poor conversions. Properly formatted epub documents are worth their weight in effort.
Epub documents _are_ HTML files in a zip archive - I'd argue that if a web site is a proper presentation of the source material, an epub is even better.
Additionally, PDF documents _are_ worse for any kind of document technical or otherwise. PDF documents are primarily Postscript instructions without any relation or hierarchy to the included elements. Epub documents/HTML provide semantic relationships and hints to the content.
Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#45Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus…
I have mixed feelings about this. I've been through nonstandard analysis. My response was, "We shouldn't need the axiom of choice to define the derivative." But I think that it is extremely important to understand that the infinitesimal notation really MEANS something. Here is some Python to demonstrate. # d is a functor. It takes a function and returns a second function. # The second function captures the change in…
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Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#46Earlier quoted context omitted.
Most calculus students don't need the full formal power of rigorous analysis. Calculus, taken alone and with the elementary properties of the real numbers assumed and a few elementary properties of infinitesimals (0 <<< infinitesimal^2 <<< infinitesimal <<< any positive real), can get you a lot of power for very little formal work.
Absolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function". You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.
It’s extremely niche (abstract, “Platonic”) to ever need to care about derivatives of non piecewise-analytic functions.
Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#47My calculus teacher tried to talk me out of taking the AP Calculus AB exam because I wasn’t doing that well in his class. “Calculus the Easy Way” which teaches in the setting of a mythical fantasy world where applied problems are solved with calculus unlocked the secret of calculus for me. I ended up scoring a “4” which was a big triumph for me, as it placed me out of taking calculus with all the people who did not w…
Douglas Downing's Trigonometry Made Easy is also a really great book for non-maths people. Same approach of situating Trigonometry in a mythical fantasy land. I remember Trigonometry as a chore of trying to memorize double-angle formulas and such, but this book really helped connect trig in an intuitive way. I didn't know he had a calculus book, I'll have to check it out now.
> Douglas Downing's Trigonometry Made Easy is also a really great book for non-maths people.
> but this book really helped connect trig in an intuitive way.
To me, understanding math in an intuitive way is actually how "maths people" think about math. The numbers and formulas are just a means to an end to get there.
Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#48Earlier quoted context omitted.
Absolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function". You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.
It’s extremely niche (abstract, “Platonic”) to ever need to care about derivatives of non piecewise-analytic functions.
I think of wavelets and stochastic processes to be a significantly bigger niche than you probably do.
Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#49Earlier quoted context omitted.
Absolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function". You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.
You're right. But then again, lots of scaffolding gets discarded when an arch gets constructed also.
Lest you disregard this as completely useless superstructure, note that basically the entirety of the theory of stochastic processes, starting with Brownian motion, is positively infested with continuous everywhere nondifferentiable functions; while the existence of a nonconstant infinitely smooth function with an identically zero Taylor series is what permits the Berezinskii-Kosterlitz-Thouless phase transition to exist. So while the weird animals of elementary real analysis are perhaps not the most important thing in the world, they are far from irrelevant to it.
Re: Calculus Made Easy by Silvanus P. Thompson (1910)
#50> What one fool can do, another can.
This quote helped me get over my fear of math. It's probably the fewest words that have made the largest impact in my life.