How much of introductory calculus can be learned without using analytic geometry or for that matter any algebraic notations but simple euclidean geometry? Are there any resources(new ones not the old ones of Newton's times) that treat calculus topics like derivative, integrals, maxima and minima using pure euclidean geometry? The genuine problems that I face is how to understand and explain polynomial curves,trigonometric and logarithmic functions and their derivatives or integrals without use of algebra.But I still believe that a high school kid will have more affinity towards the pure geometric visualization rather than the algebraic one. So do suggest some good resources(preferably books) on this kind of approch( I found one going by the name Calculus without Analytic Geometry by Barry Mitchell, but struggled to get its contents).

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    $\begingroup$ What do you mean by pure Euclidiean geometry and how do you differentiate it from analytic geometry? If you mean no coordinate system at all and not even a numerical notion of slope, then I am highly skeptical of your belief that a high school student would find the resulting theory more accessible than the standard approach. What do you base your belief on? $\endgroup$ Mar 20, 2017 at 0:47
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    $\begingroup$ Didn't Newton already do this in the Principia ? As I understand, this makes the presentation of even the simplest things in calculus virtual jibberish. Algebra is the bedrock on which calculus which us ordinary folks can master rests. To do calculus without algebra, it takes Archimedes. $\endgroup$ Mar 20, 2017 at 1:42
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    $\begingroup$ You may be interested in sections 1.1-3 of this book lightandmatter.com/fund . E.g., there is a geometrical proof (in an example, not the main text) that the derivative of an exponential is an exponential with the same base. The derivative of the log can be done in a similar way. Keisler gives a nice geometrical proof of the derivative of the sine. I don't see how you would do a general proof of $(x^n)'=nx^{n-1}$ by geometrical methods, but maybe I'm not creative enough. $\endgroup$
    – user507
    Mar 20, 2017 at 3:30
  • $\begingroup$ Not totally on your question, but my understanding is that both Seki's and Barrow's proofs were essentially "Euclidean" in the sense I think you mean. But it's not particularly useful without connecting it to some sort of coordinate system, really. $\endgroup$
    – kcrisman
    Mar 20, 2017 at 13:34
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    $\begingroup$ The creation of coordinates by Descartes was a major advance for a reason. No coordinates means no graphs of functions, a much weaker connection between geometry and algebra, and no trigonometric or logarithmic functions in any serious sense. Without a function concept you have a severely crippled conception of what calculus can do and little in the way of its computational power (e.g., change of variables in an integral). Maybe you can teach some broad notions from calculus in a very simple setting, but the student will be incapable of handling nearly any conventional task in calculus. $\endgroup$
    – KCd
    Mar 21, 2017 at 0:51

1 Answer 1


There are some calculus courses for kids that focus on the splitting up into slices and that sort of thing. They don't go as far as you'd probably like, and they aren't Euclidean, really. This is a course that was offered last year: http://naturalmath.com/2016/02/newsletter-february-1-2016/


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