Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection. H. Weyl 1952
In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. Euclide (~300 BC)
It's possible to pass only one line parallel to a given line through a point that lies outside of it.
In geometry I find certain imperfections which I hold to be the reason why this science [...] can as yet make no advance from that state in which it came to us from Euclid. As belonging to these imperfections, I consider [...] the momentous gap in the theory of parallels, to fill which all efforts of mathematicians have so far been in vain. N. Lobachevsky 1826
I have discovered such wonderful things that I was amazed...out of nothing I have created a strange new universe. — Jánus Bolyai to his father
To praise it would amount to praising myself. For the entire content of the work...coincides almost exactly with my own meditations [in the] past thirty or thirty-five years.” — Gauss to Farkas Bolyai
Manifolds in which, as in the plane and in space, the line-element may be reduced to the form,are therefore only a particular case of the manifolds to be here investigated; they require a special name, and therefore these manifolds in which the square of the line-element may be expressed as the sum of the squares of complete differentials I will call flat.” Riemann
Given a [homogeneous] manifold and a transformation group acting [transitively] on it, to investigate those properties of figures on that manifold which are invariant under transformations of that group. F. Klein 1872
Every [differentiable] symmetry of the action of a physical system [with conservative forces] has a corresponding conservation law. E. Noether 1918
Unification of electromagnetic and weak forces (modelled with the groups U(1) × SU(2)) and the strong force (based on the group SU(3)) C. N. Yang & R. L. Mills 1954
It is only slightly overstating the case to say that Physics is the study of symmetry. P. Anderson 1972