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a symbolism which we do not understand. Also a symbolism, which we only partially understand and are unaccustomed to use, is difficult to follow. In exactly the same way the technical terms of any profession or trade are incomprehensible to those who have never been trained to use them. But this is not because they are difficult in themselves. On the contrary they have invariably been introduced to make things easy. So in mathematics, granted that we are giving any serious attention to mathematical ideas, the symbolism is invariably an immense simplification. It is not only of practical use, but is of great interest. For it represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other. If any one doubts the utility of symbols, let him write out in full, without any symbol whatever, the whole meaning of the following equations which represent some of the fundamental laws of algebra:—

$x+y=y+x$

$(1)$

$(x+y)+z=x+(y+z)$

$(2)$

$x\times y=y\times x$

$(3)$

$(x\times y)\times z=x\times (y\times z)$

$(4)$

$x\times (y+x)=(x\times y)+(x\times z)$

$(5)$

Here $(1)$ and $(2)$ are called the commutative and associative laws for addition, $(3)$ and $(4)$