In George Simmons' Calculus Gems there is an interesting quote, supposedly from Galileo, pertaining to whether one can compare curved and straight lines (in length, for instance):

Who is so blind as not to see that, if there are two equal straight lines, one of which is then bent into a curve, that curve will be equal to the straight line?

In the context of the book, this seems to be in response to La Géométrie, but Simmons also likes telling good stories, so perhaps this is a separate quote - or perhaps it's apocryphal, or maybe it's just not in a well-known source.

In any case, any tips as to what dialogue/letter this is from would be greatly appreciated.

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    $\begingroup$ I do not think a "response": Descartes' Geometrie was published in 1637, only five years before Galileo death. $\endgroup$ May 14 at 19:32
  • $\begingroup$ This is a bit odd because Simmons says that the occasion was Torricelli's computation of the length of the logarithmic spiral. However, Descartes's opinion that "ratios between straight and curved lines are not known, and... cannot be discovered" concerned algebraic curves and not curves generated by motions like the spiral. He did not doubt that bending a string would preserve its length, he objected to curves obtained by such bending as "non-geometrical", see Mancosu and Crippa. $\endgroup$
    – Conifold
    May 14 at 23:01
  • $\begingroup$ @Conifold, This is a bit odd because the common wisdom seems to be that Descartes viewed only algebraic curves as the domain of geometry, and on the contrary viewed mechanically-produced ones as lying outside that domain, so it should be the opposite... Leibniz famously challenged Descartes' assumption in this area. $\endgroup$ May 15 at 15:03
  • $\begingroup$ Thanks for the clarifications on Descartes' views re: mechanical versus algebraic curves. (And I suspected as much about the timing, though Simmons is vague on the point.) So ... do people that this purported quote is apocryphal? I can't imagine Simmons inventing it completely from whole cloth, given that he does cite a fair number of his stories, but maybe something got garbled along the way? $\endgroup$
    – kcrisman
    May 20 at 12:50


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