Mémoire sur la surfaces élastiques.
[1707]. So long as algebra and geometry proceeded separately their progress was slow and their application limited, but when these two sciences were united, they mutually strengthened each other, and marched together at a rapid pace toward perfection.—Lagrange.
Leçons élémentaires sur les Mathématiques, Leçon Cinquième.
[1708]. The laws of algebra, though suggested by arithmetic, do not depend on it. They depend entirely on the conventions by which it is stated that certain modes of grouping the symbols are to be considered as identical. This assigns certain properties to the marks which form the symbols of algebra. The laws regulating the manipulation of algebraic symbols are identical with those of arithmetic. It follows that no algebraic theorem can ever contradict any result which could be arrived at by arithmetic; for the reasoning in both cases merely applies the same general laws to different classes of things. If an algebraic theorem can be interpreted in arithmetic, the corresponding arithmetical theorem is therefore true.—Whitehead, A. N.
Universal Algebra (Cambridge, 1898), p. 2.
[1709]. That a formal science like algebra, the creation of our abstract thought, should thus, in a sense, dictate the laws of its own being, is very remarkable. It has required the experience of centuries for us to realize the full force of this appeal.—Mathews, G. B.
F. Spencer: Chapters on Aims and Practice of Teaching (London, 1899), p. 184.
[1710]. The rules of algebra may be investigated by its own principles, without any aid from geometry; and although in many cases the two sciences may serve to illustrate each other, there is not now the least necessity in the more elementary parts to call in the aid of the latter in expounding the former.—Chrystal, George.
Encyclopedia Britannica, 9th Edition; Article “Algebra”