# who first defined a tangent to a circle as a line meeting it only once?

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].

• 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