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So, I was just reading about the Secant Method and came across the sentence that it:

"developed independently of Newton's method, and predates it by over 3,000 years.".

This would mean that it was first used some time around $1300$ BCE. Which was well before the developments of Cartesian co-ordinate systems and probably even before basic algebra, though they may have had some form of it.

So my question is: how could the ancient Babylonians possibly come up with a root-finding algorithm with such a limited set of mathematical tools at their disposal? Did they just do it by trial and error or can someone show me how they can derive the method as the Babylonians did?

Thanks very much,

Jam.

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    $\begingroup$ You might be interested in the 2013 paper Origin and Evolution of the Secant Method in One Dimension which relates it to a Rule of Double False Position illustrated in the 18th cen. BC Egyptian Rhind Papyrus. Personal JSTOR account (or subscription) required. $\endgroup$ – hardmath Sep 3 '16 at 18:12
  • $\begingroup$ Nice reference. Thanks for the reply, @hardmath $\endgroup$ – Jam Sep 3 '16 at 18:23
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    $\begingroup$ Working backward from locating the Wikipedia(?) article you read, I found that the citation there is for the same Am. Math Monthly article, though it links to a site that provides a much less generous "abstract" and no preview. To your underlying question, the thinking in Egypt and Babylon was geometrically motivated, in eras long pre-dating Cartesian co-ordinates and calculus. $\endgroup$ – hardmath Sep 4 '16 at 15:53
  • $\begingroup$ @hardmath I see. That seems logical. The connection with the arithmetical computations they did and the geometrical argument is quite impressive considering it seems to be only a few steps behind co-ordinate geometry $\endgroup$ – Jam Sep 4 '16 at 15:59
  • $\begingroup$ Is the secant method that much different from taking averages? If f(x) = -a and and f(y) = b, it seems one doesn't need many tools to assume f(z)= 0 for some z in (x,y) and a naive assumption that most things are linear it occur proportionally a/b of the interval. I don't know about this historically, but that doesn't seem to require much by way of "tools". I think most children can and do do this. Just without verbalizing it in mathy words. $\endgroup$ – fleablood Sep 4 '16 at 16:13

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