Arthur Cayley - Collected Quotes

"Metrical geometry is thus a part of descriptive geometry, and descriptive geometry is all geometry." (Arthur Cayley, "A Sixth Memoir upon Quantics", Philosophical Transactions of the Royal Society of London, 1859)

"[T]he notion of a negative magnitude has become quite a familiar one […] But it is far otherwise with the notion which is really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole of modern analysis and geometry, that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry: I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or inquiry. […] considering the prominent position which the notion occupies-say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it." (Arthur Cayley, [address before the meeting of the British Association at Southport] 1883)

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

"I would myself say that the purely imaginary objects are the only realities, the truest things, in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object; and if there is no conception of straightness, then it is meaningless to deny the conception of a perfectly straight line." (Arthur Cayley, [address before the meeting of the British Association at Southport] 1883)

"It is difficult to give an idea of the vast extent of modern mathematics. The word 'extent' is not the right one: I mean extent crowded with beautiful details - not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood and flower [...]" (Arthur Cayley, [address before the meeting of the British Association at Southport] 1883)

"Mathematics connect themselves on the one side with common life and the physical sciences; on the other side with philosophy, in regard to our notions of space and time, and in the questions which have arisen as to the universality and necessity of the truths of mathematics, and the foundation of our knowledge of them." (Arthur Cayley)

"Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience." (Arthur Cayley)