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From googling, it seems commonly believed that Euclid did this, but it seems nowhere in Euclid does he even state this property of a tangent line explicitly. Rather Euclid gives 4 other equivalent properties, that the line does not cross the circle, that it is perpendicular to the radius, that is a limit of secant lines, and that it makes an angle of zero with the circle, the first of which is his definition, the others being in Proposition III.16. I am wondering where the "meets only once" definition got started. I presume once it got going, and people stopped reading Euclid, (which seems to have occurred over 100 years ago), the currently popular definition took over. Perhaps I should consult Legendre or Hadamard? Thank you for any leads.

Well I have found this definition in Hadamard's lessons in plane geometry. Any earlier references?

I have also found another equivalent characterization of a tangent by Euclid, Prop. (III.36-37): A segment PX, from a point P outside a circle and meeting the circle at X, is tangent to the circle at X if and only if there exists another segment PB, meeting the circle first at A and then at B, such that (PA)(PB) = (PX)^2, [in terms either of equality squares of lengths of segments, or of equality of area of rectangles].

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I think looking for Hadamard is a little too recent. Perhaps if not Euclid my first try would be to look at the beginning of the speaking of tangents, which brings you to the Newton/Leibniz era. –  Patrick Da Silva Aug 2 '11 at 5:54
    
finding tangents is a old problem, one motivation for newton's calculus. decartes studied the problem maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/… and probably people before him... –  yoyo Aug 2 '11 at 15:20
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Some googling points to Archimedes (use ctrl-F to find his name in this article)

http://math.ucsd.edu/~ashenk/Section2_8.pdf

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Seems that crafty Archimedes had calculus up his sleeve all along. Damn shame we had to wait a quarter shy of two millenia for Newton et al to remake it. –  anon Aug 4 '11 at 18:52
    
Actually that article claims incorrectly that Euclid said this, and even cites Euclid for it. Maybe he means it follows from Euclid's definition. –  roy smith Aug 5 '11 at 5:29
    
I see he cites page one in Heath's translation. I do n ot have access to that, but I wonder what it says there. E.g. is he citing a remark of Heath, or is Heath making an interpretation, or a different translation? Does anyone have volume 2 of that Dover book? –  roy smith Aug 5 '11 at 5:34
    
He also cites Appolonius for one of Euclid's other characterizations as a line such that no other line lies between the curve and it. It makes me wonder if that author has read Euclid. –  roy smith Aug 5 '11 at 5:38
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