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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

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Existence claims as theorems became fashionable after Hilbert introduced the axiomatic method. Before that people more often talked about problems and constructions (following Euclid's, or rather Pappus's, distinction). In this case, the construction was of a curve given its curvature and torsion as functions of arclength (the "natural equations"). For the plane curves the construction (naturally, from curvature alone) appears in Euler's De constructione aequationum ope motus tractorii aliisque ad methodum tangentium inversam pertinentibus (On the construction of equations using dragged motion, and of other things pertinent to the inverse method of tangents, 1736), so McCleary ascribes the "Fundamental Theorem of Plane Curves" to him. Euler first introduces the idea of natural equation (without the name), and does the calculations for the tractrix. Later he used modified versions of natural equations (with angle of inclination instead of curvature) to show that logarithmic spiral was similar to its evolute (1740), and cycloid was congruent to it (1764), see Boyer's History of Analytic Geometry, pp.227-8. According to Gray's Modern Differential Geometry of Curves and Surfaces, natural equations were subsequently studied by Lacroix and Hill, Lacroix and the Calculus by Domigues gives a lot of background on the early history of differential geometry.

However, the case of space curves had to wait until Frenet (partly in 1847) and Serret (1851) introduced the system of equations for the moving frame, which are required for the construction. It was then accomplished by Hoppe in Ueber die Darstellung der Curven durch Krümmung und Torsion (1862). Struik does not seem to have been aware of Hoppe's work, as he attributes the Riccati equations method to Lie (1882) and Darboux (1887). Scofield in Curves of Constant Precession writes:

"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."

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