=1704.= The human mind has never invented a labor-saving machine equal to algebra.-- _The Nation, Vol. 33, p. 237._ =1705.= They that are ignorant of Algebra cannot imagine the wonders in this kind are to be done by it: and what further improvements and helps advantageous to other parts of knowledge the sagacious mind of man may yet find out, it is not easy to determine. This at least I believe, that the _ideas of quantity_ are not those alone that are capable of demonstration and knowledge; and that other, and perhaps more useful, parts of contemplation, would afford us certainty, if vices, passions, and domineering interest did not oppose and menace such endeavours. --LOCKE, JOHN. _An Essay concerning Human Understanding, Bk. 4, chap. 3, sect. 18._ =1706.= Algebra is but written geometry and geometry is but figured algebra.--GERMAIN, SOPHIE. _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._