Who first introduced the functional definition of symmetry?

Question

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!

Answer 1

If I understand correctly, you mean the definition of an object’s {symmetries} as its stabilizer under some group action on an ambient space. While a clearcut “first” might not exist, I’d say Jordan has early approximations to that, provided you are willing to replace “function” by “transformation”, “substitution”, or “motion”. E.g. in his Traité des substitutions (1870, p. 50):

§V. — Symmetry of rational functions.

The link between groups and functions.

    60. Let $\mathrm F_1$ be an arbitrary rational function of $k$ letters $a, b, c,\dots$; $\mathrm F_1, \mathrm F_\alpha, \mathrm F_\beta$ the $1.2.\dots k$ functions obtained by letting the $1.2.\dots k$ substitutions $1, \alpha, \beta,\dots$ operate on these letters (...) The $M$ substitutions $1,\alpha,\dots$ which don’t alter the function $\mathrm F_1$ obviously form a group, which one can call the function’s group.

(Earlier Serret’s Cours d’algèbre supérieure (1866, p. 387) wrote that the “substitutions admises” by a function form a “système conjugué” ($=$ Cauchy’s term for group, p. 251). I’d have thought crystallographic groups were defined as stabilizers before this, but it seems not.)

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EDIT: To answer the question as recast in your comment (“first published book or article where symmetries in geometry are defined as transformations”), I think a problem is that early sources won’t call any maps symmetries: instead you’ll find expressions like mouvements qui superposent à lui-même (Jordan 1867, p. 230), Transformationen in sich (Klein & Lie 1871; Klein 1893, p. 326), Deckbewegungen (Sohncke 1875, p. 115), Deckoperationen (Schoenflies 1891, p. 13; Curie 1894, p. 395; Love 1906, p. 147), or at best symmetry-operations (Hilton 1903, p. 32).

While these references all contain the main idea, the exact terminology you want does not seem to appear until A. Speiser, Die Theorie der Gruppen von endlicher Ordnung (2nd ed., 1927, p. 78):

Eine kongruente Abbildung des Gitters auf sich selber nennen wir eine Symmetrie des Gitters.

Interestingly, this sentence is absent in the first edition (1923, p. 53). In English, the earliest such definition I can find is by H. S. M. Coxeter in Regular skew polyhedra in three and four dimensions and their topological analogues (1937, p. 35):

We define a symmetry (or "symmetry operation") of any figure as a congruent transformation of the figure into itself (i.e., a combination of translations, rotations, and reflections).

Coxeter later repeats it in e.g. Mathematical Recreations and Essays (1939, p. 130) or Regular complex polytopes (1974, p. 2).

Answer 2

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.