George Bruce Halsted

“ The straight rays of light break against the mirror and are turned back. Our eyes give us no account of this break and turn, and so deceive us, putting before us, like the spiritualists, the effect of a fourth dimension. These are not questions which can be decided by reference to our space intuitions, for our intuitions are confined to Euclidean space, and even there are insufficient, approximative. For instance, you suppose you can imagine a curve on a plane, and so in physics curves are taken to represent functions. ”

“ Hence followed the principle of the separation of symbols of operation from those of quantity, with the "calculus of operations." The world of mind had now developed sufficiently to appreciate the definition of an algebra, though when it was first given I do not know. An algebra is an abstract science or calculus of symbols combining according to defined laws. There may be an indefinitely large number of sets of such defined laws—that is, of distinct, different, and independent algebras. ”

“ Among the first men to systematically consider symbols combining according to laws more complicated than those of natural number was Sir Rowan Hamilton. After a struggle of ten years from 1833, his genius enabled him to escape from the rut of common thought by casting away the commutative principle in multiplication, which in numbers formulates the fact that twice three gives precisely the same result as thrice two. So, in 1843, he presented to the Irish Academy the principles of the algebra of quaternions, and published an article on the subject in the "Philosophical Magazine" in 1844. ”

“ This somewhat complicated so-called axiom is only the converse or inverse of proposition seventeen, that "any two angles of a triangle are together less than two right angles," a theorem readily demonstrated from the preceding postulates and axioms. An inverse is usually exceedingly easy to prove. Then why not remove this inverse from among the postulates, place it after seventeen, and demonstrate it? This obvious way to improve on Euclid suggested itself to numerous geometers throughout the centuries. Hundreds tried it, and failed. ”

“ Modern logic will date from Boole.
But I wish to call attention to the fact that here we find the best, the most satisfactory introduction to the study of modern algebras, modern mathematics. When told that in these systems a product may not vary with each of its factors; that a product may vanish without either of its factors vanishing ”