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Tangents and circles

  1. Oct 1, 2015 #1
    I recent came across this paragraph by David Hume. Although he is considered a philosopher, he tried to make comments on math as well. I find this one interesting, but I have no idea what it means and what he is getting at. Out of pure curiosity, does anyone else know what this means?: "The angle of contact between a circle and its tangent is infinitely less than any rectilinear angle, so that as you may increase the diameter of the circle to infinity, this angle of contact becomes still less, even in infinity, and that the angle of contact between other curves and their tangents may be infinitely less than those between any circle and its tangent, and so on, infinity".
  2. jcsd
  3. Oct 1, 2015 #2


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    I think he's observing that if you look at a tangent touching a circle, you get a sort of intuitive feel of their making a small angle, which I think is the brain making a kind of practical average of the angles of the gradients of the circle over the parts of the approach to the contact point that we can make out.

    But then if we zoom in with a magnifier, that apparent angle gets smaller. The more we zoom in and magnify, the smaller it appears, and there is no theoretical limit to that shrinking, because the actual 'angle' at the contact point is zero.

    These days we have a mathematical language that enables us to understand and express that much more concisely and clearly. But we need to remember that Hume was writing not long after Newton and Leibniz, and the tools of calculus were not widely understood, and their consequences were not yet much explored.

    I don't know what he was getting at with the last bit that starts with 'and that the angle of contact between other curves...'. It's quite likely that it's just a red herring and he got confused. Hume was an unparalleled genius (IMHO) but I think we can forgive him not being an expert on calculus, given that that wasn't his forte, and the era when he was writing.
  4. Oct 3, 2015 #3
    I didn't know that the angle formed by two straight lines is always greater than that formed by two circles touching each other at one point
  5. Oct 3, 2015 #4
    I think this is the kind of writing you'd expect from people who are in the process of discovering calculus. It shows that nowadays (through huge efforts), calculus is much better understood. Basically, the thing he seems to be going for is curvature.
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