TITLE: Who first introduced the functional definition of symmetry?
QUESTION [10 upvotes]: Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves distance and $F(X) = X$).
Thanks!

REPLY [8 votes]: From the introduction of Legendre's Revolution (1794): The Definition of Symmetry in Solid Geometry, Giora Hon and Bernard R. Goldstein,
Archive for History of Exact Sciences, Vol. 59, No. 2 (January 2005), pp. 107-155.


The modern scientific concept of symmetry is, in the first place, a mathematical
    idea. It refers to an intrinsic property of a mathematical object which makes that object
    invariant under certain classes of transformations, such as rotation, reflection, inversion, or other operations. These invariant properties are the subject of group theory - a mathematical theory which explores, systématises and formalises these permanent features
    that are preserved by the object despite the transformation that it undergoes. The pervasiveness
    of symmetry in modern mathematics and science is well known, and it will be
    taken for granted here.
In antiquity the fundamental meaning of summetrian Greek was proportion which
    in mathematics also meant commensurability. Clearly, the meaning of symmetry (Latin:
    symmetrid) changed over time, and it will be the goal of this paper to describe the introduction
    of this term into solid geometry at the end of the 18th century with an entirely
    new meaning. We put aside for another occasion the aesthetic sense of summetria/symmetria
    in antiquity as well-proportioned (notably in Vitruvius's De architecture^ and
    its subsequent applications in architecture and art (and, to some extent, in scientific
    contexts).
In fact, in the period from 1794 to 1815 three scientists claimed to use symmetry in
    a new way: Adrien-Marie Legendre (1752-1833) in solid geometry (1794), Sylvestre
    François Lacroix (1765-1843) in algebra (1797), and René- Just Haüy (1743-1822) in
    crystallography (1815). In a series of papers now in preparation we will discuss the introduction
    of symmetry in algebra and in crystallography, and then argue that the modern
    usages of symmetry in all scientific domains flow from these three seminal figures.
...
In this paper we investigate Legendre's work on solid geometry where he introduced
    a new definition of symmetry that, we claim, has served as the basis for the modern
    scientific concept of symmetry.


This text seems to be an extremely thorough account of the history of the modern concept of symmetry.