# Who developed The Fundamental Theorem of Curves

In any modern differential geometry textbook (Do Carmo, for example), the fundamental theorem of curves can be found. It states that:

every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size) completely determined by its curvature and torsion

I am wondering who first formulated this theorem.

In D. J Struik's "History of Differential Geometry" he describes papers by Michel Ange Lancret (1774 – 1807). Lancret knew of first curvature and second curvature (torsion).

He wrote two mathematical papers...His first paper is of a more general nature. It contains the two fundamental quantities of the space curve, which he calls " premiere flexion " and " seconde flexion."...Curvature and torsion appear therefore as differentials, and are not written as finite quantities until CAUCHY.

LANCRET is therefore the first to take up the systematic theory of space curves after EULER, but it seems in an independent way. The line of progress goes here from CLAIRAUT via EULER and LANCRET to CAUCHY and FRENET.

I suspect that the theorem was developed by someone in the above quoted line of progression.

"Differential Geometry of Curves and Surfaces", Manfredo Do Carmo, 1976

https://en.wikipedia.org/wiki/Fundamental_theorem_of_curves

"This activity, called "solving natural equations", is generally achieved by solving Riccati equations... Although the solution generally exists, it usually cannot be obtained explicitly. Euler [6] found explicit integral formulae for plane curves (where $$\tau=0$$) through direct geometric analysis. Hoppe [9] developed a general method for solving the natural equations for space curves by solving Riccati equations through a complicated sequence of integral transformations. He digressed to obtain formulae for the tangent, normal, and binormal indicatrices for general helices and essentially for curves of constant precession. Enneper [5] obtained explicit closed-form solutions for helices on revolved conic sections through direct geometric analysis."