On Beauty: Beauty and Mathematics (-1899)

“The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. And since these (e.g. order and definiteness) are obviously causes of many things, evidently these sciences must treat this sort of causative principle also (i.e. the beautiful) as in some sense a cause.” (Aristotle, "Metaphysica", cca. 350 BC)

 “Thus, of all the honorable arts, which are carried out either naturally or proceed in imitation of nature, geometry takes the skill of reasoning as its field. It is hard at the beginning and difficult of access, delightful in its order, full of beauty, unsurpassable in its effect. For with its clear processes of reasoning it illuminates the field of rational thinking, so that it may be understood that geometry belongs to the arts or that the arts are from geometry.” (Agennius Urbicus, “Controversies about Fields”, cca. 4 century BC)

"Wherever there is number, there is beauty.”  (Proclus)

“Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures.” (John Arbuthnot, “An Essay on the Usefulness of Mathematical Learning”, 1701)

“By the word symmetry […] one thinks of an external relationship between pleasing parts of a whole; mostly the word is used to refer to parts arranged regularly against one another around a centre. We have […] observed [these parts] one after the other, not always like following like, but rather a raising up from below, a strength out of weakness, a beauty out of ordinariness.” (Johann Wolfgang von Goethe)

 “The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both.” (Henry Thoreau, “A Week on the Concord and Merrimack Rivers”, 1873) 

"As for everything else, so for a mathematical theory: beauty can be perceived but not explained." (Arthur Cayley, [President’s address] 1883)