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There are only four types of plane isometries: translations, rotations, reflections, and glide reflections. My question is: who was the first person who proved this? I asked this question personally to two historians (John Stillwell and Jeremy Gray) and none of them was able to provide an answer.

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  • $\begingroup$ Once the question about classification of isometries is asked, the answer is trivial. So the question who answered it first makes little sense. $\endgroup$ – Alexandre Eremenko Jul 11 '17 at 21:47
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    $\begingroup$ I am not sure why this is so trivial. Glide reflections did not occur to me spontaneously, and even if they did proving that these are all of the types does not seem obvious. $\endgroup$ – Conifold Jul 12 '17 at 0:11
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Coolidge (1940, p. 273) gives a proof which ends:

(...) Hence any real transformation of the Euclidean plane which keeps distances invariant is either a rotation, a translation, or the product of one of these and a reflection in a line.†


† The earliest proof I have seen for this is Chasles (1830, p. 321). I think, however, it must have been known earlier, as the corresponding problem in three dimensions, which I take up next, was solved by Euler [1776].

I’ll let you judge if you think (1830) contains the theorem... To me it rather seems that Coolidge has in mind what Chasles says while discussing (1830) in his later book (1837, pp. 548–549):

Quand une figure plane éprouve un mouvement infiniment petit dans son plan, il existe toujours un point qui, pendant ce mouvement, reste fixe ; (...)

En énonçant [ce théorème], il y a quelques années (voir 1830), nous l'avons présenté comme un cas particulier d'un théorème sur le déplacement fini quelconque d'une figure plane (...). Mais ces théorèmes dépendent eux-mêmes d'un principe encore plus général, que voici :

Si l'on conçoit, dans un plan, deux figures qui ont été primitivement la perspective l'une de l'autre, et qui se trouvent actuellement placées d'une manière quelconque l'une par rapport à l'autre ;

Chaque point de l'une des figures aura son homologue dans l'autre figure ;

Il existera généralement trois points, dans l'une des figures, qui se trouveront superposés respectivement sur leurs homologues dans la seconde figure ;

L'un de ces trois points sera toujours réel ; les deux autres pourront être imaginaires.

Il résulte de là qu'il y a aussi trois droites, dans l'une des figures, qui se trouvent superposées sur leurs homologues dans la seconde figure : ce sont les droites qui joignent deux à deux les trois points.

L'une de ces droites est toujours réelle, et les deux autres peuvent être imaginaires.

Quand les deux figures sont semblables, ce qui est un cas particulier de la perspective, deux des trois points et deux des trois droites sont toujours imaginaires ; le troisième point est réel ; la troisième droite est aussi réelle ; mais elle se trouve située à l'infini.

Cela a lieu pareillement quand les deux figures sont égales entre elles.

Ces propriétés des figures planes ont leurs analogues dans les figures à trois dimensions, pour lesquelles j'ai déjà énoncé quelques théorèmes qui se rapportent à cette théorie. (Voir 1830, p. 321.)

Indeed, Coolidge’s lines leading to the above quote are fleshing out this argument, which is about (what we would call) eigenvalues and eigenvectors of the isometry’s matrix $$ \begin{pmatrix} \cos\theta & \mp\sin\theta & a\\ \sin\theta & \pm\cos\theta & b\\ 0&0&1 \end{pmatrix}: $$

The characteristic equation here is of the third degree and, since it is real, it must have one real root at least, so that there is at least one real fixed point. As a circle goes into a circle, so the circular points at infinity are either invariant, or interchanged. Suppose that they are invariant and that we have a real finite fixed point [... then] we have a rotation about this point. If, however, the real fixed point is infinite [... then] we have a translation. When the circular points at infinity are interchanged and there is a finite fixed point, the transformation can be factored into a rotation about this point and a reflection in a line through it. If the fixed real point be infinite [... then] the transformation is the product of a translation and a reflection in a line through one of these points. Hence (...)

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  • $\begingroup$ The Euler paper has been translated here. $\endgroup$ – Marius Kempe Aug 29 '17 at 12:15
  • $\begingroup$ I suspect that you are wrong concerning the article by Chasles that Coolidge is quoting here. The reference provided by Coolidge says “Bulletin des Sciences Mathématiques de Ferrussac, vol. xxiv, 1831, 273”, whereas the article to which you kindly posted a link was published at the same journal, but appeared on vol. xiv, p. 321. $\endgroup$ – José Carlos Santos Dec 13 '17 at 16:01
  • $\begingroup$ @JoséCarlosSantos I believe xxiv is a typo for xiv, and 273 points to Coolidge’s above-reproduced page where he quotes that volume’s page 321. According to the Royal Society’s Catalogue, Chasles’ paper also appeared in Quetelet’s Correspondance 7 (1832) 352-357. $\endgroup$ – Francois Ziegler Dec 14 '17 at 19:48
  • $\begingroup$ @FrancoisZiegler Thank you very much. This was most helpful. $\endgroup$ – José Carlos Santos Dec 14 '17 at 19:52

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